Transcript
PHYSICAL CONSTANTS Acceleration due to gravity
g
9 .8 m / s
N A
Avogadro’s number
ELECTROMAGNETIC CONSTANTS WAV EL EN GT HS OF LI GH T IN A VAC UU M (m)
2
6.022 022 × × 10 10
23
molecules/ molecules /mol
Coulomb’s constant
k
9 × 10 × 10 9 N · m2 /C2
Gravitational constant
G
6.67 × 67 × 10 10
h
Planck’s constant
6.63 × 63 × 10 10
N · m2 /kg 2
11
−
34
−
Ideal gas constant
R
Permittivity of free space
ε0
8.8541 × 8541 × 10 10
Permeability of free space
µ0
4π × 10 × 10
J ·s
8.314 J/ J/(mol· (mol · K) = 0 .082 atm· atm · L/(mol· (mol · K)
7
−
12
−
C/(V· (V · m)
Wb/ Wb/(A· (A · m)
6.5 – 7.0 × 10 × 10
−
7
× 10 Orange 5.9 – 6.5 × 10
−
7
Yellow
5.7 – 5.9 × 10 × 10
−
7
Green
4.9 – 5.7 × 10 × 10
−
7
Blue
4.2 – 4.9 × 10 × 10
−
7
Violet
4.0 – 4.2 × 10 × 10
−
7
Red
ƒ = frequency (in Hz) 108
109
1010
1011
1012
radio microwav microwaves es waves 1 10-1 10-2 = wavelength (in m)
10-3
1013
1014
1015
infrar infrared ed 10-4
10-5
1016
1018
10-7 R
10-8
O
Y G
1019
20
10
gamma X rays rays
ultraviolet
10-6
10-9 B
10-10
I
10-11
10-12
V
= 780 nm visible light 360 nm
INDICES OF REFRACTION FOR COMMON SUBSTANCES SUBSTANCES ( l = 5.9 X 10
Air
1.00
Alcohol
1.36
Corn oil
1.47 1.47
Diamond Water
2.42 1.33
Glycerol
1017
–7
m)
331 m/s
Speed of sound at STP Speed of light in a vacuum
c
3.00 × 00 × 10 10 8 m/s
Electron charge
e
1.60 × 60 × 10 10
Electron volt
eV
1.6022 × 6022 × 10 10
u
Atomic mass unit
REFLECTION AND REFRACTION θincident = θ reflected c (v is the speed of light in the medium) Index of refraction n = v
C 19
−
J
1 .6606 × 6606 × 10 10 27 kg = 931. 931.5MeV 5MeV/c /c2 −
31
mp
1.6726 × 6726 × 10 10 27 kg = 1 .00728 u = 938. 938.3MeV 3MeV/c /c2
Opticall instrume Optica instrument nt Lens: Concave Convex
27
1.6750 × 6750 × 10 10 kg = 1 .008665 u 2 = 939. 939.6MeV 6MeV/c /c 5.976 976 × × 10 10 24 kg
Radius of Earth
6.378 378 × × 10 10 6 m
1
−
normal 0 2 angle of
angle of 0 reflection
'
refraction
refracted ray reflected ray
n2 n1
LENSES AND CURVED MIRRORS 1 1 1 + = p q f
−
Mass of Earth
θc = sin
Critical angle
−
angle of incidence 0 1
n1 sin θ1 = n 2 sin θ2
Snell’s Law
−
9.11 × 11 × 10 10 kg = 0 .000549 u = 0 .511 MeV/c MeV /c2
…of neutron
incident ray
Law of Reflection
me
Rest mass of electron
...of proton
19
−
OPTICS
image size q = − object size p
Focal Foc al distan distance ce f
Image distance q
Type of image
negative positive
negative (same side) nega negati tive ve (sam (samee sid side) e) positi positive ve (oppo (opposit sitee side) side)
virtual, erect virt virtua ual, l, erec erectt real, real, inver inverted ted
p < f p > f
1
p
2 3
h V
F
Mirror: Convex
negative
Concave
positive
negative (opposite side)
virtual, erect
4
negati negative ve (oppos (opposite ite side) side) posi positi tive ve (sam (samee side side))
virtua virtual, l, erect erect real real,, inve invert rted ed
5
q
p < f p > f
DYNAMICS
6
6
NEWTON’S LAWS 1. Firs Firstt Law: Law: An An object re mains in its state of rest or motion with constant velocity unless acted upon by a net external force. dp F = F = 2. Second Second Law: Law: F net net = ma dt 3. Third rd Law Law:: For every action there is an equal and opposite reaction.
Normal force
F p
1
V
V
F N = mg cos mg cos θ (θ is the angle to the horizontal)
p
q
Kinetic friction f k = µ k F N
µs is the coefficient of static friction. µk is the coefficient of kinetic friction. For a pair of materials, µk < µs .
Notation
ˆ = a xˆi + a yˆi + a z k a = a
Magnitude
+ a + a a = | = | a| = a + a + a
Dot product
= a x bx + a + a y by + a + az yz a · b = a
2 x
(θ is the angle between a and b)
|a × b | = ab = ab sin θ a × b points in the direction given by the right-hand rule:
2 y
2 z
= ab cos θ
Cross product
a x b
W = W =
1 p2 mv 2 = 2 2m
Gravitational potential energy
b
�� �� ��
5
P avg avg =
Instantaneous power
Change in velocity
P = P = F · v
MOMENTUM AND IMPULSE = m v p = m
Impulse
J = F t = ∆p J =
dt = ∆ p F dt =
COLLISIONS m 1 v1 + m + m 2 v2 = m 1 v1 + m + m2 v2 �
All collisions
1 1 1 1 2 2 m1 v12 + m2 v22 = m1 (v ( v1 ) + m2 (v (v2 ) 2 2 2 2 �
�
v1 − v − v 2 = − (v1 − v − v 2 )
aavg =
v dt
∆ v ∆t
t (s)
VELOCITY v (m/s)
+
t (s)
∆v =
CONSTANT AC CE LE RAT IO N vf = v 0 + at + at 1 vavg = (v0 + v + v f ) 2
a dt
– ACCELERATION a (m/s 2)
+
1 s = s = s 0 + v + v 0 t + at 2
�
Elastic collisions �
DISTANCE s (m)
Instantaneous d v a = acceleration dt
∆ W ∆t
Linear momentum
∆ s ∆t
vavg =
Average acceleration
Average power
F
q
Instantaneous d s v = velocity dt
U g = mgh E = = KE + U
V p
KINEMATICS
Displacement ∆s =
Total mechanical energy
h F
4
Average velocity
F · d · ds
∆ U = = −W − W
a
− a z by ) ˆi + (a ( az bx − a − a x bz ) jˆ + (a ( ax by − a − a y bx ) ˆ a × b = (ay bz − a k ax a y a z = ax a y b z ˆi ˆ jˆ k
�� �� ��
KE =
Kinetic energy
a b
F
V p q
3
(for conservative forces)
VE CT OR FO RM UL AS
h
F
W = W = F · s = F = F s cos θ
Work
Potential energy
5 9 . 3 $
q
V
q
Work-Energy Theorem W = W = ∆KE
UNIFORM CIRCULAR MOTION v 2 mv 2 Centripetal acceleration a c = Centripetal force F c = r r
F p
2
FRICTION Static friction f s, s, max = µ s F N
N A C 5 9 . 5 $
F
WO RK , EN ER GY, PO WE R
F w = mg
Weight
h
h
�
1 = s 0 − v − v f t + at 2 = s 0 + v + v avg t vf 2 = v 02 +
t (s)
–
2 a(sf − s 0 ) CONTINUED ON OTHER SIDE
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WAV ES
ELECTRICITY
Amplitude A
Frequency f Wavelength λ T =
1 f
=
Period T
2 π
Angular frequency ω
ω = 2πf =
ω
2 π
Tension in string F T
x − T t λ
mass length
Mass density µ =
Length L
F T µ
Speed of standing wave
v =
Wavelength of standing wave
λn =
2 L
f beat = |f 1 − f 2 |
DOPPLER EFFECT Motion of source Stationary Motion of observer Stationary
Toward obser ver at vs
Away from obser ver at vs
v
veff = v
veff = v
λ
λeff = λ
f
f eff = f
veff = v + vo λeff = λ f eff = f v+v v
Towards source at vo
v −v v
λeff = λ
v v −v
f eff = f
� s
s
o
λeff = λ
Away from source at vo veff = v − vo λeff = λ f eff = f v−vv
f eff = f
� o
ω =
Angular velocity
Angular acceleration
αavg =
s
∆ ω ∆t
r
v =
ωf = ω 0 + αt
R R
I =
r 2 dm
sphere
� g
v = 0 U = max KE = 0
5
U e =
v = 0 U = max KE = 0
I =
Resistance
R = ρ
Ohm s Law
I =
Power dissipated by resistor
P = V I = I 2 R
Heat energy dissipated by resistor
W = P t = I 2 Rt
L A
V R
Series circuits I eq = I 1 = I 2 = I 3 = . . . V eq = V 1 + V 2 + V 3 + · · · Req = R 1 + R 2 + R 3 + · · ·
R 1
ML
MR
T = 2π
L
rod
TORQUE AND ANGULAR MOMENTUM Torque τ = F r sin θ d L τ = r × F τ = dt τ = I α Angular momentum
L = pr sin θ
L = r × p
L = I ω KE rot =
1 Iω 2 2
2π T
=
k m
R 2 R 3
Magnetic force on moving charge
F = qv B sin θ
F = q (v × B)
Magnetic force on current-carrying wire
F = B I� sin θ
F = I ( � × B)
MAGNETIC FIELD PRODUCED BY…
P V = nRT
Combined Gas Law
P 1 V 1 P 2 V 2 = T 1 T 2
µ 0 q v × ˆ r 4π r 2
Magnetic field due to a moving charge
B =
Magnetic field produced by a current-carrying wire
B =
Magnetic field produced by a solenoid
B = µ 0 nI
Biort-Savart Law
dB =
µ0 I
2π r
µ 0 I (d� × ˆ r) r2 4π
ε = −
Lenz s Law and Faraday s Law ’
’
m k
E · dA =
1. First Law ∆ (Internal Energy) = ∆Q + ∆ W 2. Second Law: All systems tend spontaneously toward maximum entropy. ∆ Qout Alternatively, the efficiency e = 1 − ∆Qin of any heat engine always satisfies 0 ≤ e < 1 . Boyle s Law
P 1 V 1 = P 2 V 2
Charles s Law
P 1 P 2 = T 1 T 2
’
’
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Q enclosed ε0
r e l t s s e l e K i n n a i D t s t J u y t , k a , r s M e v g n o , r e w d s b o e m d v D s i e a i . d l r l i F K e m l M a l h i W l e a l . a i r n O n r a e n W . n S B A O : a s : D r s n : r o t o a n i t o d u D : i t E b n a s i r g t t i i r e n s s r u e o e l l S C D I
B · dA = 0
Gauss s Law for magnetic fields ’
s
∂ ΦB ∂ =− E · ds = − ∂t ∂t c
Faraday s Law ’
B · dA
s
B · ds = µ 0 I enclosed
Ampere s Law ’
c
B · ds = µ 0 I enclosed + µ0 ε0
Ampere-Maxwell Law
c
∂ ∂t
E · dA
s
GRAVITY m 1 m2 r2
Newton s Law of Universal Gravitation
F = G
Acceleration due to gravity
a =
Gravitational potential
U (r ) = −
Escape velocity
vescape =
’
and A = (∆x)max is the amplitude.
THERMODYNAMICS
dΦB dt
MAXWELL’S EQUATIONS
s
is the angular frequency
GAS LAWS Universal Gas Law
R 3
R 1
’
x = A sin( ωt )
Equation of motion
where ω =
R 2
Parallel circuits I eq = I 1 + I 2 + I 3 + · · · V eq = V 1 = V 2 = V 3 = . . . 1 1 1 1 = + + + ··· Req R1 R2 R2
Gauss s Law
1 k (∆x)2 2
2
disk
Rotational kinetic energy
v = max U = min KE = max
MASS-SPRING SYSTEM Restoring force F = −k (∆) x ∆x is the distance the spring is stretched or compressed from the equilibrium position, and k is the spring constant.
Period 2
MR
∆ Q ∆t
Current
MAGNETISM
mg cos 0
S T R A H C K R A P S
M T
CIRCUITS
mg sin 0
2
12
R
R
ring
W q
T
Elastic potential energy 1
2
MR
2
s
equilibrium position
MOMENTS OF INERTIA ( I )
1
∆V =
F = E q
q
m g
ωf 2 = ω 02 + 2 α(θf − θ0 )
2
Potential difference
F on q
o
0
= θ 0 + ω avg t
MR 2
v ±v v ±v
2g� (1 − cos θmax )
1 αt 2
particle
s
T = 2π
1 ωavg = (ω0 + ω f ) 2
E =
KIRCHHOFF’S RULES Loop rule: The sum of all the (signed) potential differences around any closed loop is zero. Node rule: The total current entering a juncture must equal the total current leaving the juncture.
v ±v v
�
Period
a
Moment of inertia
s
Velocity at equilibrium position
v
CONSTANT
θ = θ 0 + ω 0 t +
v v+v
PENDULUM
r dθ ω = dt a t α = r dω α = dt
∆ θ ωavg = ∆t
s
SIMPLE HARMONIC MOTION
θ =
Angular position
v+v v
�
veff = v ± vo
�
ROTATIONAL MOTION
Electric field
1 q 1 q 2 q 1 q 2 = 4πε 0 r2 r2
’
n
SOUND WAVES Beat frequency
F = k
’
� �
WAVE ON STR IN G
Coulomb s Law
T
Wave speed v = f λ Wave equation y (x, t) = A sin(kx − ωt ) = A sin 2π
ELECTROSTATICS
s r o r r e t / a m o s c r . o s r r e e t t o r n o k p r e a R p s . w w w
N A C 5 9 . 5 $ 5 9 . 3 $ 4
0 4 3 6 3 3 9 5 0 2
7
GM Earth 2 rEarth
GM m r
GM r
KEPLER’S LAWS OF PLANETARY MOTION 1. Planets revolve around the Sun in an elliptical path with the Sun at one focus. 2. The imaginary segment connecting the planet to the Sun sweeps out equal areas in equal time. 3. The square of the period of revolution is directly proportional to the cube of the length of the semimajor axis of revolution: T 2 is constant. 3 a
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