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BASIC PHYSICS 1 MODULE Raw Data · June 2016 DOI: 10.13140/RG.2.1.36 10.13140/RG.2.1.3639.9601 39.9601
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MAASAI MARA UNIVERSITY
(MMU)
SCHOOL OF SCIENCE
DEPARTMENT OF PHYSICS
PHY 110 MODULE FOR
BSC, BED, BSC (COMPUTER)
PHY110: BASIC PHYSICS 1
Maera John
JAN-APRIL 2015
Page 1 of 64
MAASAI MARA UNIVERSITY
(MMU)
SCHOOL OF SCIENCE
DEPARTMENT OF PHYSICS
PHY 110 MODULE FOR
BSC, BED, BSC (COMPUTER)
PHY110: BASIC PHYSICS 1
Maera John
JAN-APRIL 2015
Page 1 of 64
SYMBOLS USED Take Note
Further Reading
Question
Written Exercises
A question: This symbol symbol indicates that that there is a …?...
Activity
Summary
S
A Congratulations
Definitions of Key Words Words
Self-Diagnosis Test Written Assignment W r i t tt t e n e Assi gnme nt
?
100
My score Objectives
o
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Table of Contents Symbols used .................................................................................................................................................... 2 TOPIC 1:
MECHANICS ..................................................................................................................................... 6
Self diagnostic Test ........................................................................................................................................... 6 Introduction ...................................................................................................................................................... 6 Specific objectives ............................................................................................................................................. 6 Nature of physics .............................................................................................................................................. 6 Physical and non-physical quantities ................................................................................................................ 6 Vectors and scalars ........................................................................................................................................... 7 Vector addition ................................................................................................................................................. 7 Dot Product or Scalar Product .......................................................................................................................... 7 The VECTOR PRODUCT or CROSS PRODUCT ..................................................................................................... 8 Written Exercise 1.1 .......................................................................................................................................... 9 TYPES OF MOTION ............................................................................................................................................ 9 Equations of uniformly accelerated linear motion ...................................................................................... 10 Classic version ............................................................................................................................................. 10 Examples ..................................................................................................................................................... 11 Written Exercise 1.2 ........................................................................................................................................ 11 Projectile Motion ............................................................................................................................................ 12 Introduction ................................................................................................................................................ 12 The Trajectory of a Simple Projectile is a Parabola ..................................................................................... 13 Summary ..................................................................................................................................................... 13 Circular Motion ............................................................................................................................................... 14 Equations of circular motion ....................................................................................................................... 14 Simple Harmonic Motion ................................................................................................................................ 15 Hooke's Law: ............................................................................................................................................... 16 Definitions:.................................................................................................................................................. 16 Dynamics of simple harmonic motion......................................................................................................... 17 Energy of simple harmonic motion ............................................................................................................. 18 Newton's laws of motion ................................................................................................................................ 18 Impulse ........................................................................................................................................................... 19 Relationship to the conservation laws ........................................................................................................ 20 Kepler's laws ................................................................................................................................................... 20 Page 3 of 64
Examples of Kepler's Third Law ................................................................................................................... 21
Friction .............................................................................................................................................................. 21 Coulomb friction ........................................................................................................................................... 22 Coefficient of friction .................................................................................................................................. 22
Reducing friction........................................................................................................................................... 23 Summary ..................................................................................................................................................... 23 TOPIC TWO:
PROPRTIES OF MATTER ............................................................................................................... 25
Self diagnostic Test ......................................................................................................................................... 25 Introduction .................................................................................................................................................... 25 Specific objectives........................................................................................................................................... 25 Phases of matter. ............................................................................................................................................ 25 Six Common Phase Changes ....................................................................................................................... 26 Density and Elasticity ...................................................................................................................................... 26 BERNOULLI'S EQUATION: for Ideal Fluid Flow ................................................................................................ 30 Written Exercise 2.1 ........................................................................................................................................ 31 TOPIC THREE:
THERMAL PHYSICS .................................................................................................................. 33
Self diagnostic Test ......................................................................................................................................... 33 Introduction .................................................................................................................................................... 33 Specific objectives ........................................................................................................................................... 33 Rankine scale .............................................................................................................................................. 33 Thermometers ................................................................................................................................................ 34 Calibration of thermometers .......................................................................................................................... 34 Coefficient of thermal expansion .................................................................................................................... 35 Thermal expansion coefficient .................................................................................................................... 35 Linear thermal expansion ........................................................................................................................... 35 Area thermal expansion .............................................................................................................................. 36 Volumetric thermal expansion .................................................................................................................... 36 Conservation of energy ................................................................................................................................... 37 The first law of thermodynamics .................................................................................................................... 37 Heat capacity or specific heat ......................................................................................................................... 38 Kinetic theory of gases .................................................................................................................................... 41 Postulates of kinetic theory of gases .............................................................................................................. 41
Pressure ..................................................................................................................................................... 42 Page 4 of 64
Temperature and kinetic energy ................................................................................................................. 44 Number of collisions with wall .................................................................................................................... 45 RMS speeds of molecules ........................................................................................................................... 46 Heat transfer ................................................................................................................................................... 46 Conduction.................................................................................................................................................. 46 Convection .................................................................................................................................................. 47 Radiation ..................................................................................................................................................... 48 Clothing and building surfaces, and radiative transfer ................................................................................ 48 Newton's law of cooling .............................................................................................................................. 49 One dimensional application, using thermal circuits .................................................................................. 49 Insulation and radiant barriers .................................................................................................................... 51 Critical insulation thickness ......................................................................................................................... 52 Blackbody Radiation ....................................................................................................................................... 52 TOPIC FOUR:
SOUND.................................................................................................................................... 55
Self diagnostic Test ......................................................................................................................................... 55 Introduction .................................................................................................................................................... 55 Specific objectives ........................................................................................................................................... 55 Sound .............................................................................................................................................................. 55 Perception of sound ........................................................................................................................................ 55 Physics of sound.............................................................................................................................................. 56 Longitudinal and transverse waves ................................................................................................................. 56 Sound wave properties and characteristics ................................................................................................ 56 Speed of sound ........................................................................................................................................... 56 Acoustics and noise ..................................................................................................................................... 57 Sound pressure level ................................................................................................................................... 57 Examples of sound pressure and sound pressure levels ............................................................................. 57 Equipment for dealing with sound .............................................................................................................. 58 References .......................................................................................................................................................... 61 Appendix 1: COURSE outline Phy110 .............................................................................................................. 62
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:
MECHANICS
TOPIC 1
SELF DIAGNOSTIC TEST Answer all questions
100 ?
1. What is physics? 2. What is a physical quantity? 3. Distinguish between vectors and scalars giving five examples for each 4. Discuss the types of motion 5. Relate Newton’s laws to Kepler’s laws of motion
INTRODUCTION In this topic we shall discuss the physical quantities, their measurement units and classify them into scalars and vectors. Operation of vectors is core in dealing with mechanical systems. We shall then derive the equations of linear, rotational, circular motion and simple Harmonic motion. Finally we shall relate Newton’s law to Kepler’s laws.
SPECIFIC OBJECTIVES
o
At the end of this TOPIC you should be able to :
1. Distinguish between vectors and scalars 2. Add, subtract and multiply vectors 3. Derive the equations of various types of motion 4. State Newton’s laws of motion 5. State the conservation of energy and momentum
NATURE OF PHYSICS Physics, is a major science, dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. Sometimes in modern physics a more sophisticated approach is taken that incorporates elements of the three areas listed above; it relates to the laws of symmetry and conservation, such as those pertaining to energy, momentum, charge, and parity. PHYSICAL AND NON-PHYSICAL QUANTITIES Dfn 1: A physical quantity is one that can be measured and a number assigned to it. An example is
time, mass, volume, power, force, amount of substance. A laboratory is not a physical quantity but its length, width and height are physical quantities. Dfn 2: A non-physical quantity is one that cannot be measured and a number given to value. An example is smell. One can only say good, bad, average but not give a number objectively.
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VECTORS AND SCALARS Dfn: Vectors are quantities that have a size and a direction. Examples: Time, temperature, and Mass Dfn:
Scalars quantities have only size. Examples: acceleration, momentum, velocity, electric field
Vectors are central to the study of physics. Early on in this study you will encounter types of vector quantities. Besides displacement and velocity, other examples of vectors include acceleration, force, gravitational field, torque, and electric and magnetic fields. Classify the basic physical quantities into scalars and vectors?
VECTOR ADDITION
The unit vectors i, j, and k are chosen so that through the addition of multiples of themselves with each other, the three of them can describe all vectors possible in the space. An arbitrary vector V, is described by specifying the amounts of i, j, and k which, when summed together, make V. The components of Vare Vx, Vy, and Vz . To specify V, it is sufficient to specify its three components (V x, Vy, and Vz). Hence, a three dimensional vector is an ordered set of three numbers. A seventy dimensional vector is an ordered set of 70 numbers. Two vectors, A and B, are equal if and only if each of their components are equal: A x = Bx; Ay = By; Az = Bz. It is interesting to observe that 1 vector equation ( A = B) is equivalent to three scalar equations. This brevity is a nice aspect to vector algebra. When we add vectors, we add each of their components separately. By this it is clear that in order to add two vectors, they must have the same dimension - otherwise the operation is undefined. When we visualize this in space, we imagine moving the start point of one vector to the endpoint of the other vector. The sum vector is the resultant. In a two dimensional case, we have mathematically A + B = (4, 7) + (4, 1) = (8, 8). We should also remember that multiplication of a vector by scalar is multiplication of each component by the same scalar, namely cA = (cAx, cAy, cAz) The length of a vector is called its magnitude and is usually denoted by | A|. The directionality of the vector is lost for this quantity so it is a scalar.
DOT PRODUCT OR SCALAR PRODUCT One way of combining two vectors is through an operation called the dot product. It is written as: A B
B A
A B COS
Another form of the equation is A B
(Axi
Ay j
Az k) (Bxi
By j
Bzk)
A x Bx
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Ay By
Az Bz
This last form can be seen clearly when we consider the dot product of the unit vectors i, j, and k. Because they are fixed at 90 degrees from each other we have: i i
j j
k k
COS0
and
1
i j
j k
k i
COS
0 2
This property of these three vectors (the dot product with themselves produces 1 and the dot product with each other produces 0) is what defines them as being a set of orthogonal, normalized vectors or orthonormal, for short. In a 2 or 3 dimensional space, we can characterize this condition as having the vectors placed at right-angles to each other. The same concept holds in higher order spaces, but we are unable to visualize "right angles" in a 70 dimensional space! With this information it is clear to see that this provides a quick route to the magnitude of a vector, namely to take the dot product of a vector with itself. A A
A
2
Ax
2
Ay
2
Az
2
A
Ax
2
Ay
2
Az
2
1 / 2
THE VECTOR PRODUCT OR CROSS PRODUCT There is another way to combine vectors. This is the cross product but in this case the result is another vector, rather than a scalar as with the dot product. It is a little more involving mathematically to remember the form of the operation so we cast it in the form of a determinant (is that any easier to remember?)
Vector multiplication yielding another vector Yields a vector which has a direction determined by the right hand rule Yields a vector perpendicular to the plane containing the other two vectors The cross product DOES NOT commute
a × b = a1i × (b1i + b 2 j + b3k) + a 2 j× (b1 i + b 2 j + b3k) + a3k × (b1i + b 2 j + b3 k) ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
a × b = (a 2b3 - a 3b2 )i + (a3b1 - a1b3 )j + (a1b2 - a 2 b1 )k ˆ
i
ˆ
a × b = a1 b1
j
k
a2
a3
b2
b3
ˆ
ˆ
ˆ
ˆ
Take a look at the order of the subscripts in the result and you will see a cyclical appearance of each one. Learn to appreciate the order in this for it will appear time and again. There are a couple of other properties worth noting here. Page 8 of 64
i i ˆ
i
ˆ
ˆ
j
ˆ
j k ˆ
k ˆ
Using the definition of cross product and right hand rule: j j k k 0
ˆ
k ˆ
ˆ
ˆ
j i
ˆ
ˆ
ˆ
i
k
i
ˆ
j
ˆ
ˆ
ˆ
ˆ
i
ˆ
i
ˆ
k ˆ
Graphically, the concept to remember is that the cross product produces a vector which is perpendicular to both vectors making up the argument of the product. This means it is orthogonal to both (though the two argument vectors need not be orthogonal to each other). When the two original vectors are orthogonal to each other, the cross product vector has the greatest magnitude (it is at its longest). As the two vectors are rotated in towards each other, the resultant vector shortens until it disappears when the two overlap. This same happens when the two initial vectors rotate away from each other, the resultant disappearing when the point opposite each other.
WRITTEN EXERCISE 1.1 1. Write down five physical quantities and five non-physical quantities apart from the ones given above 2. Find sum, difference, scalar product and cross product of the following vectors (i) A 3i 5 j 10k and C 3 j - 2i - 6k (ii) (5i - j + 2k) and (2i + 3j- k) ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Revision Exercises 1.1 1.24
What displacement at 70° has a component of 450 m? What is its y-component?
Ans.
1.3 km, 1.2km
1.25
What displacement must be added to a 50 cm displacement in the +x-direction to give a resultant displacement of 85 cm at 25°? Ans. 45 cm at 53°
TYPES OF MOTION In physics, equations of motion describe the behavior of a system (e.g., the motion of a particle under an influence of a force) as a function of time. Sometimes the term refers to the differential equations that the system satisfies (e.g., Newton's second law), and sometimes to the solutions to those equations.
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Equations of uniformly accelerated linear motion The body is considered between two instants in time: one "initial" point and one "current". Often, problems in kinematics deal with more than two instants, and several applications of the equations are required. If acceleration, a is constant, a differential, adt , may be integrated over an interval from 0 to Δt (Δt = t − ti), to obtain a linear relationship for velocity. Integration of the velocity yields a quadratic relationship for position at the end of the interval. v
u
a t;
s
si
u t
1 2
a( t)2
and v 2
u2
2a(s
si )
where... ui ----is the body's initial velocity; si-----is the body's initial position and its current state is described by: v--- The velocity at the end of the interval s---- the position at the end of the interval (displacement) Δt--- the time interval between the initial and current states a---- the constant acceleration, or in the case of bodies moving under the influence of gravity, a = g. Note that each of the equations contains four of the five variables. Thus, in this situation it is sufficient to know three out of the five variables to calculate the remaining two. Classic version The above equations are often written in the following form: v
u
at;
s
ut
1 2
2 at ;
s
1 2
(u
v)t and v
2
u
2
2as
where s = the distance between initial and final positions (displacement) (sometimes denoted R or x) u = the initial velocity (speed in a given direction) v = the final velocity a = the constant acceleration t = the time taken to move from the initial state to the final state
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Examples Many examples in kinematics involve projectiles, for example a ball thrown upwards into the air. Given initial speed u, one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity g. At this point one must remember that while these quantities appear to be scalars, the direction of displacement, speed and acceleration is important. They could in fact be considered as uni-directional vectors. Choosing s to measure up from the ground, the acceleration a must be in fact −g, since the force of gravity acts downwards and therefore also the acceleration on the ball due to it. At the highest point, the ball will be at rest: therefore v = 0. Using the fourth equation, we have:
s
v
2
u 2g
2
Substituting and cancelling minus signs gives:
s
u
2
2g
WRITTEN EXERCISE 1.2 1. Derive the three equations of linear motion 2. How motion under gravity affect the three equations?
Revision Exercises 1.2 2.15
A ball that is thrown vertically upward on the Moon returns to its starting point in 4.0 s. The acceleration due to gravity there is 1.60 m/s2 downward. Find the ball's original speed.
2.30
A truck starts from rest and moves with a constant acceleration of 5.0 m/s2. Find its speed and the distance Ans. 20 m/s, 40 m traveled after 4.0 s has elapsed.
2.31
A box slides down an incline with uniform acceleration. It star ts from rest and attains a speed of 2.7 m/s in 3.0 Ans. (a) 0.90 m/s2; (b) 16 m s. Find (a) the acceleration and (b) the distance moved in the first 6.0 s.
2.32
A car is accelerating uniformly as it passes two check points that are 30 m apa rt. The time taken between checkpoints is 4.0 s, and the car's speed at the first checkpoint is 5.0 m/s. Find the car's acceleration and its Ans. 1.3 m/s 2, 10 m/s speed at the second checkpoint.
2.33
An auto's velocity increases uniformly from 6.0 m/s to 20 m/s while covering 70 m in a straight line. Find the Ans. 2.6 m/s 2, 5.4 s acceleration and the time taken.
Page 11 of 64
PROJECTILE MOTION Introduction A projectile is any object that is cast, fired, flung, heaved, hurled, pitched, tossed, or thrown. The path of a projectile is called its trajectory. Some examples of projectiles include …
a bullet the instant it exits the barrel of a gun or rifle a moving airplane in the air with its engines and wings disabled the space shuttle or any other spacecraft after main engine cut off (MECO)
The force of primary importance acting on a projectile is gravity. This is not to say that other forces do not exist, just that their effect is minimal in comparison. A tossed heliumfilled balloon is not normally considered a projectile as the drag and buoyant forces on it are as significant as the weight. Helium-filled balloons can't be thrown long distances and don't normally fall. In contrast, a crashing airplane would be considered a projectile. Even though the drag and buoyant forces acting on it are much greater in absolute terms than they are on the balloon, gravity is what really drives a crashing airplane. The normal amounts of drag and buoyancy just aren't large enough to save the passengers on a doomed flight from an unfortunate end. A projectile is any object with an initial non-zero, horizontal velocity whose acceleration is due to gravity alone. An essential characteristic of a projectile is that its future has already been preordained. The only relevant quantities that might vary from projectile to projectile then are initial velocity and initial position This is where we run into some linguistic complications. Airplanes, guided missiles, and rocket-propelled spacecraft are sometimes also said to follow a trajectory. Since these devices are acted upon by the lift of wings and the thrust of engines in addition to the force of gravity, they are not really projectiles. To get around this dilemma, it is common to use the term ballistic trajectory when dealing with projectiles. The laws of physics are assumed universal until it can be demonstrated otherwise. The unification of physical law is a theme that surfaces from time to time in physics. A projectile and a satellite are both governed by the same physical principles even though they have different names. A simple projectile is made mathematically simple by an idealization (basically a lie of convenience). By assuming a constant value for the acceleration due to gravity, we make the problem easier to solve and (in many cases) do not really lose all that much in the way of accuracy. Every projectile problem is essentially two one-dimensional motion problems … The kinematic equations for a simple projectile are those of an object traveling with constant horizontal velocity and constant vertical acceleration. Horizontal
Vertical
Quantity Page 12 of 64
ax = 0
ay = −g
acceleration
vx = ux
vy = uy − gt
velocity-time
x = x0 + uxt
y = y0 + uyt − ½ gt2
displacement-time
vy2 = uy2 − 2g(y − u)
velocity-displacement
The Trajectory of a Simple Projectile is a Parabola max range at 45°, equal ranges for launch angles that exceed and fall short of 45° by equal amounts (ex. 40° & 50°, 30° & 60°, 0° & 90°) Summary
S
A projectile is any object … with an initial non-zero, horizontal velocity o whose acceleration is due to gravity alone. o The path of a projectile is called its trajectory. A projectile is said to be … a simple projectile if the acceleration due to gravity may be assumed o constant in both magnitude and direction throughout its trajectory. o a satellite if it follows a closed path that never brings it in contact with a celestial body (like the earth). a general projectile no matter where its trajectory may take it. o The kinematic equations for a simple projectile are those of an object traveling with … o constant horizontal velocity and constant vertical acceleration. o he horizontal distance traveled by a projectile is called its range. projectile launched on level ground with an initial speed u at an angle θ above the horizontal … will have the same range as a projectile launched with an initial speed o u at (90° − θ). (Identical projectiles launched at complementary angles have the same range.) will have a maximum range when θ = 45°. o
Revision Exercises 1.3 2.43
A marble, rolling with speed 20 cm/s, rolls off the edge of a table that is 80 cm high, (a) How long does it
Page 13 of 64
take to drop to the floor? (b) How far, horizontally, from the table edge does the marble strike the floor? Ans. (a) 0.40 s; (b) 8.1 cm
2.44
A body projected upward from the level ground at an angle of 50° with the horizontal has an initial speed of 40 m/s. (a) How long will it take to hit the ground? (b) How far from the starting point will it strike? (c) At what angle with the horizontal will it strike? Ans. (a) 6.3 s; (b) 0.16 km; (c) 50°
2.45
A body is projected downward at an angle of 30° with the horizontal from the top of a building 170 m high. Its initial speed is 40 m/s. (a) How long will it take before striking the ground? (b) How far from the foot of the building will it strike? (c) At what angle with the horizontal will it strike? Ans. (a) 4.2 s; (b) 0.15 km; (c) 60°
CIRCULAR MOTION For circular motion at a constant speed v, the centripetal acceleration of the motion can be derived. Since in radian measure,
Equations of circular motion The analogues of the above equations can be written for rotation:
ω = ωo + αt;
θ=
1 2
(ω + ωo )t; θ = ωo t +
1 2
2
2
2
αt ; ω = ωo + 2αθ and θ = ωt -
where: α is the angular acceleration ω is the angular velocity θ is the angular displacement ω0 is the initial angular velocity.
Page 14 of 64
1 2
αt
2
Revision questions 1.4 9.19
A flywheel turns at 480 rpm. Compute the angular speed at any point on the wheel and the tangential speed 30.0 cm from the center. Ans. 50.3 rad/s, 15.1 m/s
9.20
It is desired that the outer edge of a grinding wheel 9.0 cm in radius move at a rate of 6.0 m/s. (a) Determine the angular speed of the wheel, (b) What length of thread could be wound on the rim of the wheel in 3.0 s when it is turning at this rate? Ans. (a) 67 rad/s; (b) 1 8 m
9.21
Through how many radians does a point on the Earth's surface move in 6.00 h as a result of t he Earth's rotation? What i s t he spe ed of a point on th e equat or? Take the radius of the Earth to be 6370km. Ans. 1.57 rad, 463 m/s
9.22 A wheel 25.0 cm in radius turning at 120 rpm increases its frequency to 660 rpm in 9.00 s. Find (a) the constant angular acceleration in rad/s 2 , and (b) the tangential acceleration of a point on its rim. Ans. (a) 6.28 rad/s 2 ; (b) 157 cm/s 2
SIMPLE HARMONIC MOTION Simple harmonic motion (SHM) is the motion of a simple harmonic oscillator, a motion that is
neither driven nor damped. A body in simple harmonic motion experiences a single force which is given by Hooke's law; that is, the force is directly proportional to the displacement x and points in the opposite direction. The motion is periodic: the body oscillates about an equilibrium position in a sinusoidal pattern. Each oscillation is identical, and thus the period, frequency, and amplitude of the motion are constant. If the equilibrium position is taken to be zero, the displacement x of the body at any time t is given by x(t)
A cos( 2 ft
)
Where A is the amplitude, f is the frequency, and Φ is the phase.
The frequency of the motion is determined by the intrinsic properties of the system (often the mass of the body and a force constant), while the amplitude and phase are determined by the initial conditions (displacement and velocity) of the system. The kinetic and potential energies of the system are also determined by these properties and conditions. Introduction
Simple harmonic motion showed both in real space and phase space. The orbit is periodic. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams) A typical example of a system that undergoes simple harmonic motion is an idealized spring – mass system, which is a mass attached to a spring. If the spring is outstretched, there is no net force on the mass (that is, the system is in mechanical equilibrium). However, if the mass is displaced from equilibrium, the spring will exert a restoring force, which is a force that tends to restore the mass to the equilibrium position. In the case of the spring – mass system, this force is the elastic force, which is given by Hooke's Law, Idea: Any object that is initially displaced slightly from a stable equilibrium point will
oscillate about its equilibrium position. It will, in general, experience a restoring force that depends linearly on the displacement x from equilibrium: Page 15 of 64
Hooke's Law: F s
-kx
(1)
where the equilibrium position is chosen to have x -coordinate x = 0 and k is a constant that depends on the system under consideration. The units of k are: k
Newtons
(2)
metres
Definitions:
Amplitude (A): The maximum distance that an object moves from its equilibrium
position. A simple harmonic oscillator moves back and forth between the two positions of maximum displacement, at x = A and x = - A. Period ( T ): The time that it takes for an oscillator to execute one complete cycle of its motion. If it starts at t = 0 at x = A , then it gets back to x = A after one full period at t = T . Frequency ( f ): The number of cycles (or oscillations) the object completes per unit time. f
1
T
(3)
The unit of frequency is usually taken to be 1 Hz = 1 cycle per second. Simple Harmonic Oscillator: Any object that oscillates about a stable equilibrium position and experiences a restoring force approximately described by Hooke's law. Examples of simple harmonic oscillators include: a mass attached to a spring, a molecule inside a solid, a car stuck in a ditch being ``rocked out'' and a pendulum.
Note:
The negative sign in Hooke's law ensures that the force is always opposite to the direction of the displacement and therefore back towards the equilibrium position (i.e. a restoring force). Page 16 of 64
The constant k in Hooke's law is traditionally called the spring constant for the system, even when the restoring force is not provided by a simple spring. The motion of any simple harmonic oscillator is completely characterized by two quantities: the amplitude, and the period (or frequency). Dynamics of simple harmonic motion
For oscillation in a single dimension, combining Newton's second law (F = m d 2x/dt 2) and Hooke's law (F = −kx, as above) gives the second-order differential equation
m
2 d x
dt
2
kx where
m is the mass
of the body, x is its displacement from the mean position, and k is a constant. The solutions to this differential equation are sinusoidal; one solution is x(t)
A cos( 2 ft
)
where A, ω, and Φ are constants, and the equilibrium position is chosen to be the origin. Each of these constants represents an important physical property of the motion: A is the amplitude, ω = 2πf is the angular frequency, and Φ is the phase.
Position, velocity and acceleration of an harmonic oscillator Using the techniques of differential calculus, the velocity and acceleration as a function of time can be found:
Acceleration can also be expressed as a function of displacement:
, Now since ma = −mω2x = −kx, ;
; Then since ω = 2πf,
and since T = 1/f, These equations demonstrate that period and frequency are independent of the amplitude and the initial phase of the motion. Page 17 of 64
Energy of simple harmonic motion The kinetic energy K of the system at time t is
and the potential energy is
The total mechanical energy of the system therefore has the constant value
Revision Exercises 1.5 [11.2]
A spring makes 12 vibrations in 40 s. Find the period and frequency of the vibration. [Ans: 0.30 Hz]
[11.5]
A 50-g mass vibrates in SHM at the end of a spring. The amplitude of the motion is 12 cm, and the period is 1.70 s. Find: (a) the frequency, (b) the spring constant, (c) the maximum speed of the mass, (d ) the maximum acceleration of the mass, (e) the speed when the displacement is 6.0 cm, and (f) the acceleration when x = 6.0 cm.
[11.16] In Fig. below the 2.0-kg mass is released when the spring is unstretched. Neglecting the inertia and friction of the pulley and the mass of the spring and string, find (a) the amplitude of the resulting oscillation and (b) its center or equilibrium point.
k =300
m = 2.0 kg
[11.27] Find the frequency of vibration on Mars for a simple pendulum that is 50 cm long. Objects weigh 0.40 as much on Mars as on the Earth. Ans. 0.45 Hz [11.28] A "seconds pendulum" beats seconds; that is, it takes 1 s for half a cycle, (a) What is the length of a simple "seconds pendulum" at a place where g = 9.80 m/s ? (b) What is the length there of a pendulum for which T = 1.00 s? Ans. (a) 99.3 cm; (b) 24.8 cm
NEWTON'S LAWS OF MOTION Page 18 of 64
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They
are: First law
There exists a set of inertial reference frames relative to which all particles with no net force acting on them will move without change in their velocity. This law is often simplified as "A body persists its state of rest or of uniform motion unless acted upon by an external unbalanced force." Newton's first law is often referred to as the law of inertia. Second law
Observed from an inertial reference frame, the net force on a particle is equal to the time rate of change of its linear momentum: F = d(mv)/dt. When mass is constant, this law is often stated as, "Force equals mass times acceleration ( F = ma): the net force on an object is equal to the mass of the object multiplied by its acceleration." Third law
Whenever a particle A exerts a force on another particle B, B simultaneously exerts a force on A with the same magnitude in the opposite direction. The strong form of the law further postulates that these two forces act along the same line. This law is often simplified into the sentence, "To every action there is an equal and opposite reaction." Note:
These laws describe the relationship between the forces acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first published on July 5, 1687. Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.
IMPULSE An impulse I occurs when a force F acts over an interval of time Δt, and it is given by I
t
Fdt
Since force is the time derivative of momentum, it follows that I
p
m v
This relation between impulse and momentum is closer to Newton's wording of the second law. Impulse is a concept frequently used in the analysis of collisions and impacts. According to modern ideas of how Newton was using his terminology, his is understood, in modern terms, as an equivalent of: The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.
Page 19 of 64
Relationship to the conservation laws In modern physics, the laws of conservation of momentum, energy, and angular momentum are of more general validity than Newton's laws, since they apply to both light and matter, and to both classical and non-classical physics. This can be stated simply, "Momentum, energy and angular momentum cannot be created or destroyed." Conservation of energy was discovered nearly two centuries after Newton's lifetime, the long delay occurring because of the difficulty in understanding the role of microscopic and invisible forms of energy such as heat and infra-red light. Revision Exercises 1.6 8.21
An empty 150 00- kg coal c ar i s co asting on a l evel track at 5.00 m/s. suddenly 500 0 kg of coal is dumped into it from directly above it. The coal initially has zero horizontal velocity. Find the final speed of the car. Ans. 3.75 m/s.
8.22
Sand drops at a rate of 2000 kg/min from the bottom of a hopper onto a belt conveyer moving horizontally at 250 m/min. Determine the force needed to drive the conveyer, neglecting friction. Ans. 139 N
8.26 Two bodies of masses 8 kg and 4 kg move along the x-axis in opposite directions with velocities of 11 m/s — POSITIVE ^-DIRECTION and 7 m/s — NEGATIVES-DIRECTION , respectively. They collide and stick together. Find their velocity just after collision. Ans. 5 m/s — POSITIVE S-DIRECTION 8.27 A 1200-kg gun moun ted on wheels shoots an 8.0 0-kg projec tile with a muzzle velocity of 600 m/s at an angle of 30.0° above the horizontal. Find the horizontal recoil speed of the gun. Ans. 3.46 m/s
KEPLER'S LAWS LAW 1: The orbit of a planet/comet about the Sun is an ellipse with the Sun's center of mass at one
focus
This is the equation for an ellipse:
x a
2
2
y b
2 2
1
LAW 2: A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time
Page 20 of 64
n LAW 3: The squares of the periods of the planets are proportional to the cubes of their semimajor
axes: Ta2 / Tb2 = Ra3 / Rb3
Square of any planet's orbital period (sidereal) is proportional to cube of its mean distance (semi-major axis) from Sun Mathematical statement: T = kR 3/2 , where T = sideral period, and R = semi-major axis Example - If a is measured in astronomical units (AU = semi-major axis of Earth's orbit) and sidereal period in years (Earth's sidereal period), then the constant k in mathematical expression for Kepler's third law is equal to 1, and the mathematical relation becomes T2 =k R 3 Examples of Kepler's Third Law Planet
P (yr)
a (AU)
T2
R3
Mercury
0.24
0.39
0.06
0.06
Venus
0.62
0.72
0.39
0.37
Earth
1.00
1.00
1.00
1.00
Mars
1.88
1.52
3.53
3.51
Jupiter
11.9
5.20
142
141
Saturn
29.5
9.54
870
868
Friction Friction is the force resisting the relative motion of two surfaces in contact or a surface in
contact with a fluid (e.g. air on an aircraft or water in a pipe). It is not a fundamental force, as it is derived from electromagnetic forces between atoms and electrons, and so cannot be calculated from first principles, but instead must be found empirically. When contacting surfaces move relative to each other, the friction between the two objects converts kinetic energy into thermal energy, or heat. Friction between solid objects is often referred to as dry friction or sliding friction and between a solid and a gas or liquid as fluid friction. Both of these types of friction are called kinetic friction.
Page 21 of 64
Contrary to many popular explanations, sliding friction is caused not by surface roughness but by chemical bonding between the surfaces. Surface roughness and contact area, however, do affect sliding friction for micro- and nano-scale objects where surface area forces dominate inertial forces. Internal friction is the motion-resisting force between the surfaces of the particles making up the substance. Friction should not be confused with traction. Surface area does not affect friction significantly, but in traction it is essential.
Coulomb friction Coulomb friction, named after Charles-Augustin de Coulomb, is a model to describe friction forces. It is described by the equation: F f = μF n
where
F f is either the force exerted by friction, or, in the case of equality, the maximum
possible magnitude of this force. μ is the coefficient of friction, which is an empirical property of the contacting materials, F n is the normal force exerted between the surfaces
For surfaces at rest relative to each other μ = μ s, where μs is the coefficient of static friction. This is usually larger than its kinetic counterpart. The Coulomb friction may take any value from zero up to F f , and the direction of the frictional force against a surface is
opposite to the motion that surface would experience in the absence of friction. Thus, in the static case, the frictional force is exactly what it must be in order to prevent motion between the surfaces; it balances the net force tending to cause such motion. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides a threshold value for this force, above which motion would commence. For surfaces in relative motion μ = μ k , where μk is the coefficient of kinetic friction. The Coulomb friction is equal to F f , and the frictional force on each surface is exerted in the direction opposite to its
motion relative to the other surface. Coefficient of friction The coefficient of friction (also known as the frictional coefficient) is a dimensionless scalar value which describes the ratio of the force of friction between two bodies and the force pressing them together. The coefficient of friction depends on the materials used; for example, ice on steel has a low coefficient of friction (the two materials slide past each other easily), while rubber on pavement has a high coefficient of friction (the materials do not slide past each other easily). Coefficients of friction range from near zero to greater than one – under good conditions, a tire on concrete may have a coefficient of friction of 1.7.
Page 22 of 64
Reducing friction Devices such as tires, ball bearings, air cushion or roller bearing can change sliding friction into a much smaller type of rolling friction. Many thermoplastic materials such as nylon, HDPE and PTFE are commonly used for low friction bearings. They are especially useful because the coefficient of friction falls with increasing imposed load. A common way to reduce friction is by using a lubricant, such as oil, water, or grease, which is placed between the two surfaces, often dramatically lessening the coefficient of friction. The science of friction and lubrication is called tribology. Lubricant technology is when lubricants are mixed with the application of science, especially to industrial or commercial objectives. Superlubricity, a recently-discovered effect, has been observed in graphite: it is the substantial decrease of friction between two sliding objects, approaching zero levels. A very small amount of frictional energy would still be dissipated. Lubricants to overcome friction need not always be thin, turbulent fluids or powdery solids such as graphite and talc; acoustic lubrication actually uses sound as a lubricant.
Summary We have basically tackled three major issues in this TOPIC.
S
- There are threeNewton’s laws of motion Viz: Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed. The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed. To every action there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions. There are three Kepler’s laws of motion Viz: LAW 1: The orbit of a planet/comet about the Sun
is an ellipse with the Sun's center of mass at one focus LAW 2: A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time LAW 3: The squares of the periods of the planets are proportional to the cubes of their semimajor axes:
Page 23 of 64
Note: 1. A vector quantity can not be added to scalar quantity. 2. Newton’s and Kepler’s laws are applicable
to macroscopic bodies. 3. Discuss the methods of reducing friction.
Written Assignments PHY110/1
W ri tt en Assignme nt
1.
Do the following assignment and post it to: The Head Department of PHYSICS MMU P.O. Box 861, Narok- Kenya
While driving on the interstate one day at 27.8 m/s (60.0 mph) I accidentally dropped the Encyclopedia of Physics out the window, 1.15 m above the ground. Determine the following … a. the horizontal and vertical components of the book's velocity the instant I released it b. the time the book was in the air c. the horizontal distance the book traveled before hitting the ground d. the horizontal and vertical components of the book's velocity the instant it hit the ground
Page 24 of 64
: PROPRTIES OF MATTER
TOPIC TWO
SELF DIAGNOSTIC TEST Answer all questions
?
100
1. Identify the phases of matter 2. Explain the term elasticity
3. What is Bernaulli’s equation? 4. Explain laminar and turbulent motion of a fluid
INTRODUCTION In this topic we shall discuss the properties of materials. This enables us to select materials for different purposes and to deeply understand properties of materials such as elasticity, rigidity, brittleness and ductility. We wil the consider fluid flow through pipes and relationships that the flow obeys.
SPECIFIC OBJECTIVES
o
At the end of this TOPIC you should be able to :
1. Define Hooke’s laws 2. Define elasticity, surface tension and turbulency. 3. Explain factors that afffect surface tension, viscosity and streamline flow of fluid.
What is the difference between ductile and melliable materials?
PHASES OF MATTER. There are four phases of matter. 1. 2. 3. 4.
Solid Liquid Gas Plasma
The state of matter depends on the motion of the molecules that make it up. Solids: Solids are objects that have definite shapes and volumes. The atoms or molecules are tightly
packed, so the solid keeps its shape. The arrangement of particles in a solid are in a regular, repeating pattern called a crystal. Page 25 of 64
The particles in a liquid are close together, but are able to move around more
Liquids:
freely than in a solid. Liquids have no definite shape and take on the shape of the container that they are in.
Gases: Agas does not have a definite shape or volume. The particles of a gas have much
more energy than either solids or liquids and can move around freely. Plasma is a gas-like mixture of positively and negatively charged particles. Plasma
Plasma:
is the most commonly found element in the universe, making up 99% of all matter. It is found in stars, such as the sun, and in fluorescent lighting. Plasma occurs when temperatures are high enough to cause particles to collide violently and be ripped apart into charged particles.
Six Common Phase Changes 1.
Melting- temperature at which a substance changes from solid to liquid.
2.
Freezing – temperature at which a substance changes from a liquid to a solid.
3.
Evaporation – substance changes from a liquid to a gas. (Heat of Vaporization- energy a substance must absorb in order to change from a liquid to a gas.)
4.
Condensation- substance changes from a gas or vapor to a liquid.
5.
Sublimation – substance changes from a solid to a gas or vapor without changing to a liquid first (endothermic)
6.
Deposition – substance changes directly into a solid without first changing to a liquid (exothermic)
DENSITY AND ELASTICITY THE MASS DENSITY (ρ) of a material is its mass per unit volume:
mass of body
volume of body
The SI unit for mass density is kg/m 3, although g/cm3 is also used: 1000 kg/m 3 = 1 g/cm 3. The density of water is close to 1000 kg/m 3.
THE SPECIFIC GRAVITY (sp gr) of a substance is the ratio of the density of the substance to
the density of some standard substance. The standard is usually water (at 4°C) for liquids and solids, while for gases, it is usually air. Page 26 of 64
sp gr=
standard
Since sp gr is a dimensionless ratio, it has the same value for all systems of units. ELASTICITY is the property by which a body returns to its or iginal size and shape when the forces that
deformed it are removed. THE STRESS (σ) experienced within a solid is the magnitude of the force acting (F), divided by the area
(A) over which it acts: force
stress=
area of surface on which force acts
Its SI unit is the pascal (Pa), where 1 Pa = 1 N/m 2. Thus, if a cane supports a load the stress at any point within the cane is the load divided by the cross-sectional area at that point; the narrowest regions experience the greatest stress. STRAIN (‹ε) is the fractional deformation resulting from a stress. It is measured as the ratio of the
change in some dimension of a body to the original dimension in which the change occurred.
strain=
change in dimension original dimension
Thus, the normal strain under an axial load is the change in length (AL) over the original length L 0: =
L Lo
Strain has no units because it is a ratio of like quantities. The exact definition of strain for various situations is given later.
THE ELASTIC LIMIT of a body is the smallest stress that will produce a permanent distortion in the body. When a st ress in exc ess of this limit is appli ed, the bod y will not return exactl y to its original state after the stress is removed. YOUNG'S MODULUS (Y ) or the modulus of elasticity, is defined as
modulus of elasticity=
stress
strain
The modulus has the same units as stress. A large modulus means that a large stress is required to produce a given strain - the object is rigid. Accordingly, F Y =
A L
Lo
FLo AL
Its SI unit is Pa. Unlike the constant k in Hooke's Law, the value of Y depends only on the material of the wire or rod, and not on its dimensions or configuration. Consequently, Young's modulus is an important basic measure of the mechanical behavior of materials. Page 27 of 64
THE BULK MODULUS (B) describes the volume elasticity of a material. Suppose that a
uniformly distributed compressive force acts on the surface of an obj ect and is directed perpendicul ar to th e su rf ac e at al l po in ts . Th en if F is th e fo rc e ac ti ng on an d pe rp en di cu la r to an ar ea A, we define
pressure on A =P=
F A
The SI unit for pressure is Pa. Suppose that the pressure on an object of original volume F 0 is increased by an amount A/ 3 . The pres sure increas e cause s a volume change ΔV, where ΔV will be n egat ive. We t hen defi ne volume stress=ΔP and volume strain=-
V
Vo
Then Bulk modulus =
stress
and then B = -
strain
P V
Vo P
V o
V
The minus sign is used so as to cancel the negative numerical value of ΔF and thereby make B
a positive number. The bulk modulus has the units of pressure. The reciprocal of the bulk modulus is called the compressibility K of the substance. THE SHEAR MODULUS (S) describes the shape elasticity of a material. Suppose, as
shown in Fig. 12-1, that equal and opposite tangential forces F act on a rectangular block. These shearin g f orces distort t he block as indicat ed, but i ts volume remains unchanged. We define
F
s
s
A
shearing strain= ;
L Lo
distancesheared distance between surfaces
Then shear modulus
shearing modulus=
;
stress strain
F
or S
FLo A L AL Lo
Definition Of Mechanical Properties
Tensile Strength
Compressive Strength
This is the ability of a material to withstand tensile loads without rupture when the material is in tension This is the ability of a material to withstand Compressive (squeezing) loads without being crushed when the material is in compression . Page 28 of 64
Shear Strength
Toughness
Elasticity
Plasticity
Ductility
Malleability
Fatigue Strength
Hardness
This is the ability of a material to withstand offset or traverse loads without rupture occurring . This is the ability of a material to withstand shatter. A material which easily shatters is brittle. Toughness indicates the ability of a material to absorb energy This is the ability of a material to deform under load and return to its original size and shape when the load is removed. The property is required for springs This is the property of a material to deform permanently under the application of a load. Plastacine is plastic. This is the exact opposite to elasticity. This is ability of a material to stretch under the application of tensile load and retain the deformed shape on the removal of the load. A ductile material combines the properties of plasticiy and tensile strength. All materials which are formed by drawing are required to be ductile This is the property of a material to deform permanently under the application of a compressive load. A material which is forged to its final shape is required to be malleable. This is the property of a material to withstand continuously varying and alternating loads This is the property of a material to withstand indentation and surface abrasion by another hard object. It is an indication of the wear resistance of a material.e.g Diamonds are very hard.
Revision Exercises 2.1 12.31 A load of 50 kg is applied to the low er end of a st eel rod 80 cm lon g an d 0.60 cm in dia meter. How much will the rod stretch? Y = 190 GPa for steel. Ans. 73 μ m 12.32 A pl at fo rm is su sp en de d by fo ur wi re s at its co rn er s. Th e wi re s ar e 3. 0 m lo ng an d ha ve a di am et er of 2.0 mm. Young's modulus for the materi al of the wires is 180 GPa. How far will the pl atform drop (due to elongation of the wires) if a 50-kg load is placed at the center of the platform? An s. 0.65 mm 12.33 Determine the fractional change in volume as the pressure of the atmosphere (1 x 105 Pa) around a metal block is reduced to zero by placing the block in vacuum. The bulk modulus for the metal 125 GPa. Ans. 8 x 10-7
is
12.34 Compute the volume change of a solid copper cube, 40 mm on each edge, when subjected to a pressure of 20 MPa. The bulk modulus for copper is 125 GPa.
Page 29 of 64
An s.
-10mm 3
BERNOULLI'S EQUATION: FOR IDEAL FLUID FLOW Assumptions:
The fluid is incompressible and nonviscous. There is no energy loss due to friction between the fluid and the wall of the pipe. There is no heat energy transferred across the boundaries of the pipe to the fluid as either a heat gain or loss. There are no pumps in the section of pipe under consideration. The fluid flow is laminar and steady state. p1
1 2
2 1v1
1gh1
1
p2
2 2v 2
2
2gh 2
Bernoulli's Equation is basically a statement of the conservation of energy per unit volume along the pipe.
Energy Density or Energy per unit Volume (SI: J/m 3):
E V
For an ideal fluid flow the energy density is the same at all locations along the pipe. This is the same as saying that the energy of a unit mass of the fluid does not change as it flow through the pipe system. Cons tan t
P
1 2
v2
gh
* A compressed fluid or gas has the ability to do work if it is allowed to expand, i.e. it has
stored energy. The magnitude of the pressure P is equal to the Potential Energy per unit volume due to the Hydrostatic Pressure in the fluid. Note that the unit of pressure even can be expressed at a unit of energy density, Pa = N/m 2=(N. m)/(m2 . m) = J/m3. Page 30 of 64
* The kinetic energy density can be though of a pressure exerted by the fluid due to its
motion. * We have already seen that gravitational potential energy density, ρgh, is just the pressure of a fluid due to its weight.
WRITTEN EXERCISE 2.1 1. What are the factors which affect surface tension of a fluid? 2. Explain why tree branches bend towards the road on a busy tarmac road used by motor vehicles 3. Explain the equation of continuity..
Revision Questions (fluid at rest) 2.2 13.42
A spring, which may be either bronze (sp gr 8.8) or brass (sp gr 8.4), has a mass of 1.26 g when measured in air and 1.11 g in water. Which is it? Ans. brass
13.43
What fraction of the volume of a piece of quartz (p = 2.65 g/cm3) will be submerged when it is floating in a c ontainer of mercury (p = 13.6 g/cm3)? Ans. 0.195
13.47 A cube of wood floating in water supports a 200-g mass resting on the center of its top face. When the mass isremoved, the cube rises 2.00 cm. Determine the volume of the cube. Ans: 1000cm3
Revision Questions (fluid in motion) 2.3 14.29 Calculate the theoretical velocity of efflux of water from an aperture that is 8.0 m below the surface of water in a large tank, if an added pressure of 140 k Pa is applied to the surface of the water. Ans. 21 m/s
14.30
What horsepower is required to force 8.0 m3 of water per minute into a water main at a pressure of 220 kPa? Ans. 39 hp
14.31 A pump lifts water at the rate of 9.0 li ters/s from a l ake through a 5.0 cm i.d. Pipe and discharges it into the air at a point 16 m above the level of the water in the lake. What are the theoretical (a) velocity of the water at the point of discharge and (b) power delivered by the pump. Ans . (a) 4.6 m/s; (b) 2.0 hp 14.32 Water flows steadily through a horizontal pipe of varying cross-section. At one place the pressure is 130 kPa and the speed is 0.60 m/s. Determine the pressure at another place in the same pipe where the speed is 9.0 m/ s. Ans . 90 kPa.
Page 31 of 64
Summary
We have basically tackled three major issues in this TOPIC. There are four phases of matter: Solid , Liquid ,
S
Gas and Plasma The
mass density (ρ) of a material is its mass per unit volume: mass of body
volume of body
Young's modulus (Y )
or the modulus of elasticity, is
defined as
modulus of elasticity=
stress
strain
Bernoulli's Equation is basically a statement of the conservation of energy per unit volume along the pipe.
- The specific gravity (sp gr) of a substance is the ratio of the density of the substance to the density of some standard substance. the standard is usually water (at 4°c) for liquids and solids, while for gases, it is usually air - Strain has no units because it is a ratio of like
Note:
quantities. the exact definition of strain for various situations is given later.
Written Assignments PHY110/1 Do the following revision exercices and post it to:
W ri tt en Assi gnme nt
The Head Department of PHYSICS MMU P.O. Box 861, Narok- Kenya
Page 32 of 64
:
TOPIC THREE
THERMAL PHYSICS
SELF DIAGNOSTIC TEST Answer all questions
?
100
1. arrange the metals you know in order of their expansivities. 2. Show how you can convert temperature from celcius to Rankin scale 3. State the assumptions of the kinetic theory of gases 4. What is the difference between radiation and conduction methods of heat transfers?
INTRODUCTION Thermal expansion, the general increase in the volume of a material as its temperature is increased. It is usually expressed as a fractional change in length or volume per unit temperature change; a linear expansion coefficient is usually employed in describing the expansion of a solid, while a volume expansion coefficient is more useful for a liquid or a gas. If a crystalline solid is isometric (has the same structural configuration throughout), the expansion will be uniform in all dimensions of the crystal. If it is not isometric, there may be different expansion coefficients for different crystallographic directions, and the crystal will change shape as the temperature changes. In a solid or liquid, there is a dynamic balance between the cohesive forces holding the atoms or molecules together and the conditions created by temperature; higher temperatures imply greater distance between atoms. Different materials have different bonding forces and therefore different expansion coefficients.
SPECIFIC OBJECTIVES
o
At the end of this TOPIC you should be able to :
1. Describe expansion in matter 2. Discuss construction of temperature scales 3.Distinguish between C p and Cv 4.Explain the mechanisms of heat transfer
Rankine scale Rankine is a thermodynamic (absolute) temperature scale named after the Scottish engineer and physicist William John Macquorn Rankine, who proposed it in 1859. The symbol is °R). As with the Kelvin scale (symbol: K), zero on the Rankine scale is absolute zero. But the Rankine degree is defined as equal to one degree Fahrenheit, rather than the one degree Celsius used by the Kelvin scale. A temperature of 459.67°R is precisely equal to 0°F. Rankine temperature conversion formulas To find
Fahrenheit
From
Formula
Rankine
°F = °R − 459.67
Page 33 of 64
Rankine
Fahrenheit
°R = °F + 459.67
kelvin
Rankine
K = °R ÷ 1.8
Rankine
kelvin
°R = K × 1.8
Celsius
Rankine
°C = (°R ÷ 1.8) – 273.15
Rankine
Celsius
°R = (°C + 273.15) × 1.8
For temperature intervals rather than specific temperatures, 1 °F = 1 °R and 1 kelvin = 1.8 °R
THERMOMETERS A thermometer is a device that measures temperature or temperature gradient using a variety of different principles. A thermometer has two important elements: the temperature sensor (e.g. the bulb on a mercury thermometer) in which some physical change occurs with temperature, plus some means of converting this physical change into a value (e.g. the scale on a mercury thermometer). Thermometers increasingly use electronic means to provide a digital display or input to a computer. Thermometers can be divided into two separate groups according to the level of knowledge about the physical basis of the underlying thermodynamic laws and quantities. For primary thermometers the measured property of matter is known so well that temperature can be calculated without any unknown quantities. Secondary thermometers are most widely used because of their convenience. Also,
they are often much more sensitive than primary ones. For secondary thermometers knowledge of the measured property is not sufficient to allow direct calculation of temperature. They have to be calibrated against a primary thermometer at least at one temperature or at a number of fixed temperatures. Such fixed points, for example, triple points and superconducting transitions, occur reproducibly at the same temperature. There is an absolute thermodynamic temperature scale. Internationally agreed temperature scales are designed to approximate this closely, based on fixed points and interpolating thermometers. The most recent official temperature scale is the International Temperature Scale of 1990. It extends from 0.65 K (−272.5 °C; −458.5 °F) to approximately 1,358 K (1,085 °C; 1,985 °F).
CALIBRATION OF THERMOMETERS Thermometers can be calibrated either by comparing them with other certified thermometers or by checking them against known fixed points on the temperature scale. The best known of these fixed points are the melting and boiling points of pure water. (Note that the boiling point of water varies with pressure, so this must be controlled.)
Page 34 of 64
The traditional method of putting a scale on a liquid-in glass or liquid-in-metal thermometer was in three stages: 1.Immerse the sensing portion in a stirred mixture of pure ice and water and mark the point indicated when it had come to thermal equilibrium. 2.Immerse the sensing portion in a steam bath at 1 standard atmosphere (101.3 kPa; 760.0 mmHg) and again mark the point indicated. 3.Divide the distance between these marks into equal portions according to the temperature scale being used.
COEFFICIENT OF THERMAL EXPANSION When the temperature of a substance changes, the energy that is stored in the intermolecular bonds between atoms changes. When the stored energy increases, so does the length of the molecular bonds. As a result, solids typically expand in response to heating and contract on cooling; this dimensional response to temperature change is Material Properties expressed by its coefficient of thermal expansion. Different coefficients of thermal expansion can be defined for a substance depending on whether the expansion is measured by:
linear thermal expansion area thermal expansion volumetric thermal expansion
Specific heat
Compressibility
These characteristics are closely related. The volumetric thermal expansion coefficient can be defined for both liquids and solids. The linear thermal expansion can only be defined for solids, and is common in engineering applications.
Thermal expansion
Some substances expand when cooled, such as freezing water, so they have negative thermal expansion coefficients. Thermal expansion coefficient The thermal expansion coefficient is a thermodynamic property of a substance. It relates the change in temperature to the change in a material's linear dimensions. It is the fractional change in length per degree of temperature change.
dL = L0 x ( alpha x dT )
where
is the original length,
the new length, and
Linear thermal expansion
Page 35 of 64
the temperature.
The linear thermal expansion is the one-dimensional length change with temperature. Area thermal expansion The change in area with temperature can be written:
For exactly isotropic materials, the area thermal expansion coefficient is very closely approximated as twice the linear coefficient.
Volumetric thermal expansion The change in volume with temperature can be written .
The volumetric thermal expansion coefficient can be written
where is the temperature, is the volume, is the density, derivatives are taken at constant pressure ; measures the fractional change in density as temperature increases at constant pressure. For exactly isotropic materials, the volumetric thermal expansion coefficient is very closely approximated as three times the linear coefficient.
Proof:
This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, one-third of the volumetric expansion is in a single axis (a very close approximation for small differential changes). Note that the partial derivative of volume with respect to length as shown in the above equation is exact, however, in Page 36 of 64
practice it is important to note that the differential change in volume is only valid for small changes in volume (i.e., the expression is not linear). As the change in temperature increases, and as the value for the linear coefficient of thermal expansion increases, the error in this formula also increases. For non-negligible changes in volume:
Note that this equation contains the main term, , but also shows a secondary term that scales as , which shows that a large change in temperature can overshadow a small value for the linear coefficient of thermal expansion. Although the coefficient of linear thermal expansion can be quite small, when combined with a large change in temperature the differential change in length can become large enough that this factor needs to be considered. The last term, is vanishingly small, and is almost universally ignored.
CONSERVATION OF ENERGY The law of conservation of energy states that the total amount of energy in a closed system remains constant. A consequence of this law is that energy cannot be created nor destroyed. The only thing that can happen with energy in a closed system is that it can change form, for instance kinetic energy can become thermal energy. Albert Einstein's theory of relativity shows that energy can be converted to mass (rest mass) and mass converted to energy. Therefore, neither mass nor pure energy are conserved separately, as it was understood in pre-relativistic physics. Today, cons ervation of “energy” refers to the conservation of the total mass-energy, which includes energy of the rest mass. Therefore, in an isolated system, mass and "pure energy" can be converted to one another, but the total amount of energy (which includes the energy of the mass of the system) remains constant. Another consequence of this law is that perpetual motion machines can only work perpetually if they deliver no energy to their surroundings. If such machines produce more energy than is put into them, they must lose mass and thus eventually disappear over perpetual time, and are therefore impossible.
THE FIRST LAW OF THERMODYNAMICS For a thermodynamic system with a fixed number of particles, the first law of thermodynamics may be stated as: , or equivalently,
,
where δQ is the amount of energy added to the system by a heating process, δW is the amount of
energy lost by the system due to work done by the system on its surroundings and dU is the change in the internal energy of the system. The δ's before the heat and work terms are used to indicate that they describe an
increment of energy which is to be interpreted somewhat differently than the dU increment of internal energy. Work and heat are processes which add or subtract energy, while the internal energy U is a particular form of energy associated with the system. Thus the term "heat energy" for δQ means "that amount of energy added as the result of heating" rather than referring to a particular form of energy. Likewise, the term "work energy" for δW means "that amount of energy lost as the result of work". The most significant result of this distinction is the fact that one can clearly state the amount of Page 37 of 64
internal energy possessed by a thermodynamic system, but one cannot tell how much energy has flowed into or out of the system as a result of its being heated or cooled, nor as the result of work being performed on or by the system. In simple terms, this means that energy cannot be created or destroyed, only converted from one form to another. For a simple compressible system, the work performed by the system may be written , where P is the pressure and dV is a small change in the volume of the system, each of which are system variables. The heat energy may be written , where T is the temperature and dS is a small change in the entropy of the system. Temperature and entropy are also system variables.
HEAT CAPACITY OR SPECIFIC HEAT
Suppose that a body absorbs an amount of heat ΔQ and its temperature consequently rises byΔT. The usual definition of the heat capacity, or specific heat, of the body is Q
C
T
If the body consists of moles of some substance then the molar specific heat (i.e., the specific heat of one mole of this substance ) is defined c
1
Q
v
T
In writing the above expressions, we have tacitly assumed that the specific heat of a body is independent of its temperature. In general, this is not true. We can overcome this problem by only allowing the body in question to absorb a very small amount of heat, so that its temperature only rises slightly, and its specific heat remains approximately constant. In the limit as the amount of absorbed heat becomes infinitesimal, we obtain c
1
Q
v
T
In classical thermodynamics, it is usual to define two specific heats. Firstly, the molar specific heat at constant volume, denoted cv
1
dQ
v dT V
and, secondly, the molar specific heat at constant pressure, denoted cp
1
dQ
v dT P
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Consider the molar specific heat at constant volume of an ideal gas. Since , no work is done by the gas, and the first law of thermodynamics reduces to
It follows that
Now, for an ideal gas the internal energy is volume independent. Thus, the above expression implies that the specific heat at constant volume is also volume independent. Since is a function only of , we can write
The previous two expressions can be combined to give
for an ideal gas. Let us now consider the molar specific heat at constant pressure of an ideal gas. In general, if the pressure is kept constant then the volume changes, and so the gas does work on its environment. According to the first law of thermodynamics,
The equation of state of an ideal gas tells us that if the volume changes by , the temperature changes by , and the pressure remains constant, then
The previous two equations can be combined to give
Now, by definition
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so we obtain for an ideal gas. This is a very famous result. Note that at constant volume all of the heat absorbed by the gas goes into increasing its internal energy, and, hence, its temperature, whereas at constant pressure some of the absorbed heat is used to do work on the environment as the volume increases. This means that, in the latter case, less heat is available to increase the temperature of the gas. Thus, we expect the specific heat at constant pressure to exceed that at constant volume, as indicated by the above formula. The ratio of the two specific heats
cp cv
is conventionally denoted γ. We have
for an ideal gas. In fact, γ is very easy to measure because the speed of sound in an ideal gas is written
where ρ is the density. Table below lists some experimental measurements of c v and γ for common gases. The extent of the agreement between γ calculated andexperimental γ is quite remarkable. Table 2: Specific heats of common gases in joules/mole/deg. (at 15 C and 1 atm.) From Reif.
Gas
Symbol (experiment)
(experiment)
(theory)
Helium
He
12.5
1.666
1.666
Argon
Ar
12.5
1.666
1.666
Nitrogen
20.6
1.405
1.407
Oxygen
21.1
1.396
1.397
Carbon Dioxide
28.2
1.302
1.298
Ethane
39.3
1.220
1.214
Revision Questions 3.1 15.13 15.14
Compute the increase in length of 50 m of copper wire when its temperature changes from 12 °C to 32 °C. For copper, a = 1.7 x 10 -3 0 C.
Ans.
1.7cm
15.15 A rod 3.0 m long is fou nd to have expanded 0.091 cm in length after a tempe rature rise of 60 °C. Wh at is a for
the material of the rod? Ans. 5.1 x 10-6 °C
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15.15 At 15.0 °C, a bare wheel has a diameter of 30.000 cm, and the inside diameter of a steel rim is 29.930 cm. To what temperature must the rim be heated so as to slip over the wheel? For this type of steel, a= l.l0x -5 °C Ans 221 oC 15.19 Calculate the increase in volume of 100 cm3 of mercury when its temperature changes from 10 °C to 35 °C. Ans. 0.45 cm . The volume coefficient of expansion of mercury is 0.000 18 °C-1 15.23 The density of gold is 19.30 g/cm3 at 20.0 °C, and the coefficient of linear expansion is 14.3 x Compute the density of gold at 90.0°C. Ans. 19.2 g/cm3
KINETIC THEORY OF GASES
The temperature of an ideal monatomic gas is a measure related to the average kinetic energy of its atoms as they move. In this animation, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. These room-temperature atoms have a certain, average speed (slowed down here two trillion fold). Kinetic theory (or kinetic theory of gases) attempts to explain macroscopic properties of gases, such
as pressure, temperature, or volume, by considering their molecular composition and motion. Essentially, the theory posits that pressure is due not to static repulsion between molecules, as was Isaac Newton's conjecture, but due to collisions between molecules moving at different velocities. Kinetic theory is also known as the kinetic-molecular theory (of gases) or the collision theory.
POSTULATES OF KINETIC THEORY OF GASES The theory for ideal gases makes the following assumptions:
The gas consists of very small particles, all with non-zero mass. The number of molecules is large such that statistical treatment can be applied. These molecules are in constant, random motion. The rapidly moving particles constantly collide with the walls of the container. The collisions of gas particles with the walls of the container holding them are perfectly elastic. The interactions among molecules are negligible. They exert no forces on one another except during collisions. Page 41 of 64
The total volume of the individual gas molecules added up is negligible compared to the volume of the container. This is equivalent to stating that the average distance separating the gas particles is large compared to their size. The molecules are perfectly spherical in shape, and elastic in nature. The average kinetic energy of the gas particles depends only on the temperature of the system. Relativistic effects are negligible. Quantum-mechanical effects are negligible. This means that the inter-particle distance is much larger than the thermal de Broglie wavelength and the molecules are treated as classical objects. The time during collision of molecule with the container's wall is negligible as comparable to the time between successive collisions. The equations of motion of the molecules are time-reversible.
Pressure Pressure is explained by kinetic theory as arising from the force exerted by gas molecules impacting on the walls of the container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall is:
where vx is the x-component of the initial velocity of the particle. The particle impacts the wall once every 2l/vx time units (where l is the length of the container). Although the particle impacts a side wall once every 1l/v x time units, only the momentum change on one wall is considered so that the particle produces a momentum change on a particular wall once every 2l/v x time units.
The force due to this particle is:
The total force acting on the wall is:
where the summation is over all the gas molecules in the container. Page 42 of 64
The magnitude of the velocity for each particle will follow:
Now considering the total force acting on all six walls, adding the contributions from each direction we have:
where the factor of two arises from now considering both walls in a given direction. Assuming there are a large number of particles moving sufficiently randomly, the force on each of the walls will be approximately the same and now considering the force on only one wall we have:
The quantity can be written as , where the bar denotes an average, in this case an average over all particles and where N is the number of particles in the box. This quantity is also denoted by where vrms is the root-mean-square velocity of the collection of particles. Thus the force can be written as:
Pressure, which is force per unit area, of the gas can then be written as:
where A is the area of the wall of which the force exerted on is considered. Thus, as cross-sectional area multiplied by length is equal to volume, we have the following expression for the pressure
where V is the volume. Then we have
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As Nm is the total mass of the gas, the density is mass divided by volume
Nm V
. Then the
pressure is
This result is interesting and significant, because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule
1
mv
2
2
rms
which is a microscopic
property. Note that the product of pressure and volume is simply two thirds of the total kinetic energy. Temperature and kinetic energy From the ideal gas law (1) where
is the Boltzmann constant, and
the absolute temperature,
and from the above result
we have then the temperature
takes the form (2)
which leads to the expression of the kinetic energy of a molecule
The kinetic energy of the system is N time that of a molecule
K
1 2
2 Nmv rms
The temperature becomes (3) Page 44 of 64
Eq.(3)1 is one important result of the kinetic theory: The average molecular kinetic energy is proportional to the absolute temperature. From Eq.(1) and Eq.(3) 1, we have (4) Thus, the product of pressure and volume per mole is proportional to the average (translational) molecular kinetic energy. Eq.(1) and Eq.(4) are called the "classical results", which could also be derived from statistical mechanics; . Since there are degrees of freedom (dofs) in a monoatomic-gas system with kinetic energy per dof is
particles, the
(5) In the kinetic energy per dof, the constant of proportionality of temperature is 1/2 times Boltzmann constant. This result is related to the equipartition theorem. As noted in the article on heat capacity, diatomic gases should have 7 degrees of freedom, but the lighter gases act as if they have only 5. Thus the kinetic energy per kelvin (monatomic ideal gas) is:
per mole: 12.47 J
per molecule: 20.7 yJ = 129 μeV
At standard temperature (273.15 K), we get:
per mole: 3406 J per molecule: 5.65 zJ = 35.2 meV Number of collisions with wall
One can calculate the number of atomic or molecular collisions with a wall of a container per unit area per unit time. Assuming an ideal gas, a derivation results in an equation for total number of collisions per unit time per area:
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RMS speeds of molecules From the kinetic energy formula it can be shown that
with v in m/s, T in kelvins, and R is the gas constant. The molar mass is given as kg/mol. The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1% (distribution of speeds). Revision questions 3.2 17.13
Find the mass of a neon atom. The atomic mass of neon is 20.2 kg/kmol. Ans. 3.36 x 10
kg
17.14 A typical polymer molecule in polyethylene might have a molecular mass of 15 x 103. (a) What is the mass in kilograms of such a molecule? (b) How many such molecules would make up 2 g of polymer? Ans. (a) 7.5 x 10^23 kg; (b) 8 x 1019
17.15
A certain strain of tob acco mosaic virus has M = 4.0 x 107 kg/kmol. How many molecules of the virus are present in 1.0 mL of a solution that contains 0.10 mg of virus per mL? Ans. 1.5 x 1012
17.16 17.17
The pressure of helium gas in a tube is 0.200 mmHg. If the temperature of the gas is 20 °C, what is the density of the gas? (Use M He = 4.0 kg/kmol.) Ans. 4.4 x 10 -5 kg/m3 17.18 At what temp erat ure will the molecules of an ideal gas have twice the rms speed they have at 20 °C? Ans. 1170 K « 900 °C
HEAT TRANSFER Heat transfer is the transition of thermal energy from a hotter object to a cooler object
("object" in this sense designating a complex collection of particles which is capable of storing energy in many different ways). When an object or fluid is at a different temperature than its surroundings or another object, transfer of thermal energy, also known as heat transfer, or heat exchange, occurs in such a way that the body and the surroundings reach thermal equilibrium. Heat transfer always occurs from a higher-temperature object to a cooler temperature one as described by the second law of thermodynamics or the Clausius statement. Where there is a temperature difference between objects in proximity, heat transfer between them can never be stopped; it can only be slowed. Conduction Conduction is the transfer of heat by direct contact of particles of matter. The transfer of energy could be primarily by elastic impact as in fluids or by free electron diffusion as predominant in metals or phonon vibration as predominant in insulators. In other words, heat is transferred by conduction when adjacent atoms vibrate against one another, or as electrons move from atom to atom. Conduction is greater in solids, where atoms are in constant contact. In liquids (except liquid metals) and gases, the molecules are usually further apart, giving a lower chance of molecules colliding and passing on thermal energy.
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Heat conduction is directly analogous to diffusion of particles into a fluid, in the situation where there are no fluid currents. This type of heat diffusion differs from mass diffusion in behaviour, only in as much as it can occur in solids, whereas mass diffusion is mostly limited to fluids. Metals (eg. copper, platinum, gold, iron, etc.) are usually the best conductors of thermal energy. This is due to the way that metals are chemically bonded: metallic bonds (as opposed to covalent or ionic bonds) have free-moving electrons which are able to transfer thermal energy rapidly through the metal. As density decreases so does conduction. Therefore, fluids (and especially gases) are less conductive. This is due to the large distance between atoms in a gas: fewer collisions between atoms means less conduction. Conductivity of gases increases with temperature. Conductivity increases with increasing pressure from vacuum up to a critical point that the density of the gas is such that molecules of the gas may be expected to collide with each other before they transfer heat from one surface to another. After this point in density, conductivity increases only slightly with increasing pressure and density. To quantify the ease with which a particular medium conducts, engineers employ the thermal conductivity, also known as the conductivity constant or conduction coefficient, k. In thermal conductivity k is defined as "the quantity of heat, Q, transmitted in time (t) through a thickness (L), in a direction normal to a surface of area (A), due to a tem perature difference (ΔT). Thermal conductivity is a material property that is primarily dependent on the medium's phase, temperature, density, and molecular bonding. A heat pipe is a passive device that is constructed in such a way that it acts as though it has extremely high thermal conductivity. This mode of analysis has been applied to forensic sciences to analyse the time of death of humans. Also it can be applied to HVAC (heating, ventilating and air-conditioning, or building climate control), to ensure more nearly instantaneous effects of a change in comfort level setting. Convection Convection is the transfer of heat energy between a solid surface and the nearby liquid or gas in motion. As fluid motion goes more quickly the convective heat transfer increases. The presence of bulk motion of fluid enhances the heat transfer between the solid surface and the fluid. There are two types of Convective Heat Transfer:
Natural Convection: is when the fluid motion is caused by buoyancy forces that result from the density variations due to variations of temperature in the fluid. For example in the absence of a external source when the mass of the fluid is in contact with the hot surface its molecules separate and scatter causing the mass of fluid to become less dense. When this happens, the fluid is displaced vertically or horizontally while the cooler fluid gets denser and the fluid sinks. Thus the hotter volume transfers heat towards the cooler volume of that fluid. Forced Convection: is when the fluid is forced to flow over the surface by external source such as fans and pumps. It creates an artificially induced convection current.
Internal and external flow can also classify convection. Internal flow occurs when the fluid is enclosed by a solid boundary such as a flow through a pipe. An external flow occurs when Page 47 of 64
the fluid extends indefinitely without encountering a solid surface. Both these convections, either natural or forced, can be internal or external as they are independent of each other. The formula for Rate of Convective Heat Transfer. q = hA(Ts − T b) A is the surface area of heat transfer. Ts is the surface temperature and while T b is the temperature of the fluid at bulk temperature. However T b varies with each situation and is the temperature of the fluid “far” away from the surface. The h is the constant heat transfer
coefficient which depends upon physical properties of the fluid such as temperature and the physical situation in which convection occurs. Therefore, the heat transfer coefficient must be derived or found experimentally for every system analyzed. Formulae and correlations are available in many references to calculate heat transfer coefficients for typical configurations and fluids. For laminar flows the heat transfer coefficient is rather low compared to the turbulent flows, this is due to turbulent flows having a thinner stagnant fluid film layer on heat transfer surface. Radiation Radiation is the transfer of heat energy through empty space. All objects with a temperature above absolute zero radiate energy at a rate equal to their emissivity multiplied by the rate at which energy would radiate from them if they were a black body. No medium is necessary for radiation to occur; radiation works even in and through a perfect vacuum. The energy from the Sun travels through the vacuum of space before warming the earth. Both reflectivity and emissivity of all bodies is wavelength dependent. The temperature determines the wavelength distribution of the electromagnetic radiation as limited in intensity by Planck’s law of black -body radiation. For any body the reflectivity depends on the wavelength distribution of incoming electromagnetic radiation and therefore the temperature of the source of the radiation. The emissivity depends on the wave length distribution and therefore the temperature of the body itself. For example, fresh snow, which is highly reflective to visible light, (reflectivity about 0.90) appears white due to reflecting sunlight with a peak energy wavelength of about 0.5 micrometres. Its emissivity, however, at a temperature of about -5C, peak energy wavelength of about 12 micrometres, is 0.99. Gases absorb and emit energy in characteristic wavelength patterns that are different for each gas. Visible light is simply another form of electromagnetic radiation with a shorter wavelength (and therefore a higher frequency) than infrared radiation. The difference between visible light and the radiation from objects at conventional temperatures is a factor of about 20 in frequency and wavelength; the two kinds of emission are simply different "colours" of electromagnetic radiation. Clothing and building surfaces, and radiative transfer Lighter colours and also whites and metallic substances absorb less illuminating light, and thus heat up less; but otherwise colour makes little difference as regards heat transfer between an object at everyday temperatures and its surroundings, since the dominant emitted wavelengths are nowhere near the visible spectrum, but rather in the far infrared. Emissivities at those wavelengths have little to do with Page 48 of 64
visual emissivities (visible (visible colours); in the far infrared, most objects have high emissivities e missivities.. Thus, except in sunlight, the colour of clothing c lothing makes little difference as regards warmth; likewise, paint colour of houses makes little difference to warmth except when the painted part is sunlit. The main exception to this is shiny metal surfaces, which have low emissivities both in the visible wavelengths and in the far infrared. Such surfaces can be used to reduce heat transfer in both directions; an example of this is the multi-layer insulation insulation used to insulate spacecraft. Low-emissivity windows in houses are a more complicated technology, since they must have low emissivity at thermal wavelengths while remaining transparent to visible light. Newton's law of cooling A related principle, Newton's law of cooling, states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings, surroundings, or environment. The law is
Q = Thermal energy in joules h = Heat transfer coefficient A = Surface area of the heat being transferred T0 = Temperature of the object's surface Tenv = Temperature of the environment This form of heat loss principle is sometimes not very precise; an accurate formulation may require analysis of heat flow, based on the (transient) heat transfer equation in a nonhomogeneous, nonhomogeneous, or else poorly conductive, conductive, medium. An analog for continuous continuous gradients gradients is Fourier's Law. In such cases, the entire body is treated as lumped capacitance heat reservoir, with total heat content which is proportional to simple total heat capacity C , and T, the temperature of the body, or Q = C T. From the definition of heat capacity C comes the relation C = dQ/dT. Differentiating Differentiating this equation with regard to time gives the identity (valid so long as temperatures in the object are uniform at any given time): dQ/dt = C (dT/dt). This expression may be used to replace dQ/dt in the first equation which begins this section, above. Then, if T(t) is the temperature of such a body at time t , and Tenv is the temperature of the environment around the body:
where r = hA/C is a positive constant characteristic of the system, which must be in units of 1/time, and is therefore sometimes expressed in terms of a characteristic time constant t0 given by: r = 1/t0 = ΔT/[dT/dt] . Thus, in thermal systems, t0 = C/hA. (The total heat capacity C of a system may be further represented by its mass-specific heat capacity cp multiplied by its mass m, so that the time constant t0 is also given by mcp /hA). One dimensional application, using thermal circuits A very useful concept used in heat transfer applications is the representation of thermal transfer by what is known as thermal circuits. A thermal circuit is the representation of the Page 49 of 64
resistance to heat flow as though it were an electric resistor. The heat transferred is analogous to the current and the thermal resistance is analogous to the electric resistor. The value of the thermal resistance for the different modes of heat transfer are calculated as a s the denominators denominators of the developed equations. The thermal resistances of the different modes of heat transfer are used in analyzing combined modes of heat transfer. The equations describing the three heat transfer modes and a nd their thermal resistances, as discussed previously are summarized in the the table below:
In cases where there is heat transfer through different different media (for example through a composite), the equivalent resistance is the sum of the resistances of the components that make up the composite. Likely, in cases where there are different heat transfer modes, the total resistance is the sum of the resistances of the different modes. Using the thermal circuit concept, the amount of heat transferred through any medium is the quotient of the temperature change and the total thermal resistance of the medium. As an example, consider a composite wall of cross- sectional area A. The composite is made of an L 1 long cement plaster with a thermal thermal coefficient k 1 and L2 long paper faced fiber glass, with thermal coefficient k 2. The left surface of the wall is at T i and exposed to air with a convective coefficient of h i. The Right surface of the wall is at T o and exposed to air with convective coefficient h o.
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Using the thermal resistance concept heat flow through the composite is as follows:
Insulation and radiant barriers Thermal insulators are materials specifically designed to reduce the flow of heat by b y limiting conduction, conduction, convection, or both. Radiant barriers are materials which reflect radiation and therefore reduce the flow of heat from radiation sources. Good insulators are not necessarily good radiant barriers, and vice vice versa. Metal, for instance, is an excellent reflector and poor insulator. insulator. The effectiveness of an insulator is indicated by its R- (resistance) value. The R-value of a material is the inverse of the conduction coefficient (k) multiplied by the thickness (d) of the insulator. The units of resistance value are in SI units: (K· m²/W)
; Rigid fiberglass, a common insulation insulation material, has an R-value R -value of 4 per inch, while poured concrete, a poor insulator, has an R-value of 0.08 per inch. The effectiveness of a radiant barrier is indicated by its reflectivity, which is the fraction of radiation reflected. A material with a high reflectivity (at a given wavelength) has a low emissivity (at that same wavelength), and vice versa (at any specific wavelength, Page 51 of 64
reflectivity = 1 - emissivity). An ideal radiant barrier would have a reflectivity of 1 and
would therefore reflect 100% of incoming radiation. Vacuum bottles (Dewars) are 'silvered' to approach this. In space vacuum, satellites use multi-layer insulation which consists of many layers of aluminized (shiny) mylar to greatly reduce radiation heat transfer and control satellite temperature. Critical insulation thickness To reduce the rate of heat transfer, one would add insulating materials i.e with low thermal conductivity (k). The smaller the k value, the larger the corresponding thermal resistance (R) value. The units of thermal conductivity(k) are W· m -1·K -1 (watts per meter per kelvin), therefore increasing width of insulation (x meters) decreases the k term and as discussed increases resistance. This follows logic as increased resistance would be created with increased conduction path (x). However, adding this layer of insulation also has the potential of increasing the surface area and hence thermal convection area (A). An obvious example is a cylindrical pipe:
As insulation gets thicker, outer radius increases and therefore surface area increases. The point where the added resistance of increasing insulation width becomes overshadowed by the effects of surface are is called the critical insulation thickness. In simple cylindrical pipes:
BLACKBODY RADIATION "Blackbody radiation" or "cavity radiation" refers to an object or system which absorbs all radiation incident upon it and re-radiates energy which is characteristic of this radiating system only, not dependent upon the type of radiation which is incident upon it. The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity which is radiating.
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The amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range. The best of classical physics suggested that all modes had an equal chance of being produced, and that the number of modes went up proportional to the square of the frequency.
But the predicted continual increase in radiated energy with frequency (dubbed the "ultraviolet catastrophe") did not happen. Nature knew better.
Explain why water has the highest heat capacity?
1. What is black body radiation? 2. Convert 456oR to oC 3. Why are heat and work not thermodynamic quantity?
Revision Questions 3.3 19.10
A single-thickness glass window on a house actually has layers of stagnant air on its two surfaces. But if it did not, how much heat would flow out of an 80 cm x 40 cm x 3.0 mm window each hour on a day when the outside temperature was precisely 0°C and the inside temperature was 18°C? For glass, k T is 0.84 W/K-m. An s. 1.4 x 10 3 kcal/h
19.11
How many grams of water at 100 °C can be evaporated per hour per cm2 by the heat transmitted through a steel plate 0.20 cm thick, if the temperature difference between the plate faces is 100 °C? For steel, k T is 42 W/K-m. An s. 0.33 kg/h-cm 2
19.15
A sphere of 3.0 cm radius acts like a blackbody. It is in equilibrium with its surroundings and absorbs 30 kW of power radiated to it from the surroundings. What is the temperature of the sphere? Ans. 2.6 x 103 K
19.16
A 2.0 cm thick brass plate (k T = 105 W/K-m) is sealed to a glass sheet (k T = 0.80 W/K-m), and both have the same area. The exposed face of the brass plate is at 80 °C, while the exposed face of the glass is at 20 °C. How thick is the glass if the glass-brass interface is at 65 °C? Ans. 0.46 mm
Summary
We have basically tackled three major issues in this TOPIC.
S
Thermometers have been built which utilise a range of physical effects to measure temperature. Most thermometers are originally calibrated to a constant-volume gas thermometer. , or equivalently, ,
Note:
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In a Liquid in glass thermometer, the liquid expands more thean the glass.
Written Assignments PHY110/1
W ri t te n Assignme nt
Do the following assignment and post it to: The Head Department of PHYSICS MMU P.O. Box 861, Narok- Kenya
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TOPIC FOUR:
SOUND
SELF DIAGNOSTIC TEST Answer all questions
?
100
1. Distinguish between sound and light waves 2. What is ultrasonic? 3. Identify three characteristics of sound waves.
INTRODUCTION SPECIFIC OBJECTIVES
o
At the end of this TOPIC you should be able to :
1. State the properties of sound 2. Distinguish between sound and light waves. 3. Explain the characteristics of sound waves.
SOUND Sound is a travelling wave which is an oscillation of pressure transmitted through a solid, liquid, or
gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.
PERCEPTION OF SOUND For humans, hearing is normally limited to frequencies between about 12 Hz and 20,000 Hz (20 kHz), although these limits are not definite. The upper limit generally decreases with age. Other species have a different range of hearing. For example, dogs can perceive vibrations higher than 20 kHz. As a signal perceived by one of the major senses, sound is used by many species for detecting danger, navigation, predation, and communication. Earth's atmosphere, water, and virtually any physical phenomenon, such as fire, rain, wind, surf, or earthquake, produces (and is characterized by) its unique sounds. Many species, such as frogs, birds, marine and terrestrial mammals, have also developed special organs to produce sound. In some species, these have evolved to produce song and speech. Furthermore, humans have developed culture and technology (such as music, telephone and radio) that allows them to generate, record, transmit, and broadcast sound.
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PHYSICS OF SOUND The mechanical vibrations that can be interpreted as sound are able to travel through all forms of matter: gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium. Sound cannot travel through vacuum.
LONGITUDINAL AND TRANSVERSE WAVES
Sinusoidal waves of various frequencies; the bottom waves have higher frequencies than those above. The horizontal axis represents time.
Sound is transmitted through gases, plasma, and liquids as longitudinal waves, also called compression waves. Through solids, however, it can be transmitted as both longitudinal and transverse waves. Longitudinal sound waves are waves of alternating pressure deviations from the equilibrium pressure, causing local regions of compression and rarefaction, while transverse waves (in solids) are waves of alternating shear stress at right angle to the direction of propagation. Matter in the medium is periodically displaced by a sound wave, and thus oscillates. The energy carried by the sound wave converts back and forth between the potential energy of the extra compression (in case of longitudinal waves) or lateral displacement strain (in case of transverse waves) of the matter and the kinetic energy of the oscillations of the medium. Sound wave properties and characteristics Sound waves are characterized by the generic properties of waves, which are frequency, wavelength, period, amplitude, intensity, speed, and direction (sometimes speed and direction are combined as a velocity vector, or wavelength and direction are combined as a wave vector). Transverse waves, also known as shear waves, have an additional property of polarization. Sound characteristics can depend on the type of sound waves (longitudinal versus transverse) as well as on the physical properties of the transmission medium . Speed of sound The speed of sound depends on the medium through which the waves are passing, and is often quoted as a fundamental property of the material. In general, the speed of sound is proportional to the square root of the ratio of the elastic modulus (stiffness) of the medium to its density. Those physical properties and the speed of sound change with ambient conditions. For example, the speed of sound in gases depends on temperature. In 20 °C (68 °F) air at the sea level, the speed of sound is approximately 343 m/s (1,230 km/h; Page 56 of 64
767 mph) using the formula "v = (331 + 0.6T) m/s". In fresh water, also at 20 °C, the speed of sound is approximately 1,482 m/s (5,335 km/h; 3,315 mph). In steel, the speed of sound is about 5,960 m/s (21,460 km/h; 13,330 mph). The speed of sound is also slightly sensitive (a second-order anharmonic effect) to the sound amplitude, which means that there are nonlinear propagation effects, such as the production of harmonics and mixed tones not present in the original sound . Acoustics and noise The scientific study of the propagation, absorption, and reflection of sound waves is called acoustics. Noise is a term often used to refer to an unwanted sound. In science and engineering, noise is an undesirable component that obscures a wanted signal. Sound pressure level Sound pressure is defined as the difference between the average local pressure of the medium outside of the sound wave in which it is traveling through (at a given point and a given time) and the pressure found within the sound wave itself within that same medium. A square of this difference (i.e. a square of the deviation from the equilibrium pressure) is usually averaged over time and/or space, and a square root of such average is taken to obtain a root mean square (RMS) value. For example, 1 Pa RMS sound pressure (94 dBSPL) in atmospheric air implies that the actual pressure in the sound wave oscillates between (1 atm Pa) and (1 atm Pa), that is between 101323.6 and 101326.4 Pa. Such a tiny (relative to atmospheric) variation in air pressure at an audio frequency will be perceived as quite a deafening sound, and can cause hearing damage, according to the table below. As the human ear can detect sounds with a very wide range of amplitudes, sound pressure is often measured as a level on a logarithmic decibel scale. The sound pressure level (SPL) or L p is defined as
where p is the root-mean-square sound pressure and p ref is a reference sound pressure. Commonly used reference sound pressures, defined in the standard ANSI S1.1-1994, are 20 µPa in air and 1 µPa in water. Without a specified reference sound pressure, a value expressed in decibels cannot represent a sound pressure level. Since the human ear does not have a flat spectral response, sound pressures are often frequency weighted so that the measured level will match perceived levels more closely. The International Electrotechnical Commission (IEC) has defined several weighting schemes. A-weighting attempts to match the response of the human ear to noise and A-weighted sound pressure levels are labeled dBA. C-weighting is used to measure peak levels. Examples of sound pressure and sound pressure levels Source of sound
RMS sound pressure
sound pressure level
Pa
dB re 20 µPa
Theoretical limit for undistorted sound at
101,325 Page 57 of 64
191
1 atmosphere environmental pressure 1883 Krakatoa eruption Stun grenades rocket launch equipment acoustic tests threshold of pain 100 hearing damage during short-term effect 20 jet engine, 100 m distant 6 – 200 jackhammer, 1 m distant / discotheque 2 hearing damage from long-term exposure 0.6 traffic noise on major road, 10 m distant 0.2 – 0.6 moving automobile, 10 m distant 0.02 – 0.2 TV set – typical home level, 1 m distant 0.02 normal talking, 1 m distant 0.002 – 0.02 very calm room 0.0002 – 0.0006 quiet rustling leaves, calm human breathing 0.00006 auditory threshold at 2 kHz – undamaged human ears 0.00002
approx 180 at 100 miles 170-180 approx. 165 134 approx. 120 110 – 140 approx. 100 approx. 85 80 – 90 60 – 80 approx. 60 40 – 60 20 – 30 10 0
Equipment for dealing with sound Equipment for generating or using sound includes musical instruments, hearing aids, sonar systems and sound reproduction and broadcasting equipment. Many of these use electroacoustic transducers such as microphones and loudspeakers. Vibrations of frequencies greater than the upper limit of the audible range for humans — that is, greater than about 20 kilohertz. The term sonic is applied to ultrasound waves of very high amplitudes. Hypersound, sometimes called praetersound or microsound, is sound waves of frequencies greater than 10 13 hertz. At such high frequencies it is very difficult for a sound wave to propagate efficiently; indeed, above a frequency of about 1.25 × 10 13 hertz, it is impossible for longitudinal waves to propagate at all, even in a liquid or a solid, because the molecules of the material in which the waves are traveling cannot pass the vibration along rapidly enough. Many animals have the ability to hear sounds in the human ultrasonic frequency range. Some ranges of hearing for mammals and insects are compared with those of humans in the Table. A presumed sensitivity of roaches and rodents to frequencies in the 40 kilohertz region has led to the manufacture of “ pest controllers” that emit loud sounds in that frequency range to drive the pests away, but they do not appear to work as advertised. Frequency range of hearing for humans and selected animals animal frequency (hertz) low high Page 58 of 64
humans 20 cats 100 dogs 40 horses 31 elephants 16 cattle 16 bats 1,000 grasshoppers and locusts 100 rodents 1,000 whales and dolphins 70 seals and sea lions 200
20,000 32,000 46,000 40,000 12,000 40,000 150,000 50,000 100,000 150,000 55,000
What do you think the term noise means?
How do ultrasonic sensors service the marketplace? A: Ultrasonic sensors service the market by providing a cost effective sensing method with unique
properties not possessed by other sensing technologies. By using a wide variety of ultrasonic transducers and several different frequency ranges, an ultrasonic sensor can be designed to solve many application problems that are cost prohibitive or simply cannot be solved by other sensors.
Long range detection: In industrial sensing, more and more applications require detection
over distance. Ultrasonic sensors detect over long ranges up to forty feet, while limit switches and inductive sensors do not. Broad area detection: While some photo electric sensors can detect over long distances they lack the ability to detect over a wide area without using a large number of sensors. The advantage of Migatron's ultrasonic sensors is that both wide and narrow areas can be covered. All it takes is the proper ultrasonic transducer selection. Widest range of target materials: Only ultrasonic sensors are impervious to target material composition. The target material can be clear, solid, liquid, porous, soft, wood and any color because all can be detected. Non contact distance measuring: Because sound can be timed from when it leaves the transducer to when it returns, distance measuring is easy and accurate to .05% of range which equates to +or- .002 of an inch at a distance of 4 inches.
What are the advantages of ultrasonics? A: When used for sensing functions, the ultrasonic method has unique advantages over conventional
sensors...
Measures and detects distances to moving objects. Impervious to target materials, surface and color. Solid-state units have virtually unlimited, maintenance-free lifespan. Page 59 of 64
Detects small objects over long operating distances. Resistant to external disturbances such as vibration, infrared radiation, ambient noise and EMI radiation. Ultrasonic sensors are not affected by dust, dirt or high-moisture environments. State threeof sound waves Why is sound heard over corners more than light waves? A modern mehod of controlling moscuitoes in the house is production of vibrations from a coil. Explain howthis is possible? 4. Explain three applications of sound waves. 1. 2. 3.
Revision Questions 4.1 23.23
What is the speed of sound in air when the air temperature is 31 °C?
Ans.
0.35 km/s
23.24 A shell fired at a target 800 m distant was heard to strike it 5.0 s after leaving the gun. Compute the average horizontal velocity of the shell. The air temperature is 20 °C. Ans. 0.30 km/s
23.29 At S.T.P., the speed of sound in air is 331 m/s. Determine the speed of sound in hydrogen at S.T.P. if the specific gravity of hydrogen relative to air is 0.0690 and if 7 = 1.40 for both gases. Ans . 1.26 km/s 23.30
Helium is a monatomic gas that has a density of 0.179 kg/m3 at a pressure of 76.0 cm of mercury and a temperature of precisely 0 °C. Find the speed of compression waves (sound) in helium at this temperature and pressure. An s. 970 m/s.
23.34 A sound has an intensity of 5.0 x 10 7 W/m2. What is its intensity level? Ans.
57 dB
23.35 A person riding a power mower may be subjected to a sound of intensity 2.00 x 10-2 W/m2. What is the intensity level to which the person is subjected? Ans. 103 dB 23.36 A rock band might easily produce a sound level of 107 dB in a room. To two significant figures, what is the sound intensity at 107 dB? Ans. 0.0500 W/m2
Summary
S
We have basically tackled three major issues in this TOPIC.
The scientific study of the propagation, absorption, and reflection of sound waves is called acoustics. Noise is a term often used to refer to an unwanted sound. In science and engineering, noise is an undesirable component that obscures a wanted signal. Sound waves are characterized by the generic properties of waves, which are frequency, wavelength, period, amplitude, intensity, speed, and direction . Sound is a travelling wave which is an oscillation of pressure transmitted through a solid, liquid, or Page 60 of 64
gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.
Note: Sound is a longitudinal wave while light is transverse wave.
Written Assignments PHY110/1
W ri t te n Assi gnme nt
Do the following assignment and post it to: The Head Department of PHYSICS MMU P.O. Box 861, Narok- Kenya
References 1. 2.
Paul A., Tipler; Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1 (6th ed.). New York, NY: Worth Publishers. pp. 666 –670. ISBN 1-4292-0132-0. W. Murray Bullis (1990). "Chapter 6". In O'Mara, William C.; Herring, Robert B.; Hunt, Le e P.. Handbook of semiconductor silicon technology . Park Ridge, New Jersey: Noyes Publications. p. 431. ISBN 0-81 55-
3. Halliday, David; Resnick, Robert; Walker, Jearl (2004-06-16). Fundamentals of Physics (7 Sub ed.). Wiley. ISBN 0471232319. 4. Hanrahan, Val; Porkess, R (2003). Additional Mathematics for OCR. London: Hodder & Stoughton. p. 219. ISBN 0-340-86960-7. 5. YoungD.H & R.A Freedman (1998) International students Edtn University physics with modern th physics 9 edtn. Addison-Wesley ltd.USA. 6. Raymond A. Serway and J.W. Jewett (2006) Physics for scientists and Engineers 6th edition. Thomson.USA 7. Fredrick J.B, Eugene H. (1998) Theory and Problems in college Physics 9th edition.Schaum’s series. NY. USA. 8. Nelkon, M. (1978)Mechanics and Properties of Matter. (5 th Ed.) London: 9. KIE (2008). Secondary school physics books 1-4. Jomo Kenyatta Foundation. Nairobi.
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APPENDIX 1: COURSE OUTLINE PHY110
MAASAI MARA UNIVERSITY School of Science Department of PHYSICS
Course:
PHY 110: BASIC PHYSICS 1
Lecture: Maera John (Mr)
Credit: 4
E-mail :
[email protected]. Cell phone: +254-722-325306
Consultation hrs: _Tues 2-4pm and Frid 2-5pm____Office No: PHY LAB Purpose:
To introduce the student to basic concepts in physics Objectives:
At the end of the course the learner should be able to: i) Resolve vectors ii) Describe the various types of motion iii) Explain the properties of matter, iv) State and derive Newton’s and Kepler’s laws of motion, v) Discuss expansionof matter and heat transfer mechanisms, vi) Explain temperature scales and thermometers, vii) Explain the kinetic theory of gases, viii) Discuss the velocity of sound through media and ix) Discuss ultrasonics and its aaplications,
COURSE OUTLINE Week one, two and Three: Mechanics - Physical and non-physical quantities - Vectors: classification; addition and subtraction; scalar product; vector product - Types of motion: Linear; Projectile; Circular; Simple Harmonic motion(SMH) - Derivation of equations of motion of each kind and applications - Newton’s laws of motion - Conservation of Energy and momentum - Friction: causes and its prevention Week four, five and six: Properties of M atter - Elasticity - Surface tension - Viscosity
-
Fluid flow
Week seven: CAT one - Sit in CAT one - Revision of CAT one Week Eight, nine and ten: Thermal physics
-
Expansion of solids, liquids and gases Temperature scales
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- Thermometers - First law of thermodynamics - Specific heat capacities of gases - Kinetic theory of gases - Mechanism of heat transfer (conduction, convection and Radiation) Week Eleven and twelve: Sound -
Characteristics of sound
-
Ulatrsonics and its applications
Genaral equation of a wave Velocity of sound in medium Waves on a string Velocity and elasticity of medium
Week Thirteen:
-
Sit in CAT two Revision of CAT two
Week fourteen and fifteen: End of semester Examinations - Individual learner revision - Sit-in examinations Course assessment:
ASSIGNMENTS: 5%
PRACTICALS: 10%
CATS: 15% EXAM: 70%.
Assessment schedule First test: Second Test:
________7 th week ________ 13 th week
References
YoungD.H & R.A Freedman (1998) Raymond A. Serway and J.W. Jewett (2006) Fredrick J.B, Eugene H. (1998)
Nelkon, M. (1978) KIE (2008).
International students Edtn University physics with modern physics 9th edtn. Addison-Wesley ltd.USA. Physics for scientists and Engineers 6th edition. Thomson.USA Theory and Problems in college Physics 9th edition.Schaum’s series. NY. USA. Mechanics and Properties of Matter. (5 th Ed.) London: Secondary school physics books 1-4. Jomo Kenyatta Foundation. Nairobi. End
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