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 �
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
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