Physics 130 Formula Sheet
Created using LATEX by Benjamin Kong
1. Pre-Midterm
Driven Oscillators
Simple Harmonic Motion
mẍ =
Occurs when F / x
r
2⇡
m
= 2⇡
T =
!
r k r
k
k1 + k2 + ...
! = 2⇡f =
=
m
m
F =
kx = ma
ẍ =
!2 x
number, d the separation of the sources, and
bẋ + F0 cos (!t)
ẍ + 2⇣!0 ẋ + !02 x =
F0
cos !t
m
L the path length di↵erence.
For constructive interference,
!0 = fnatural
! = fdriven
✓
◆
2!0 !⇣
= tan 1
!02 ! 2
F0
1
q
A=
m (! 2 ! 2 )2 + 4! 2 ! 2 ⇣ 2
0
multiple of 2⇡: sin ✓ =
⇡
!A sin (!t + ),
ahead of x
2
0
Max v at equilibrium position.
a(t) = ! 2 A cos (!t + ), s
⇡ ahead of x
p
v2
v = ! A2 x2 , Amax = x20 + 02
!
✓
◆
v0
= tan 1
, check w/ CAST!
!x0
ET = 0.5mv 2 + 0.5kx2 = 0.5kA2
p
Beats: !
¯=
1
fbeat = |f1
Small Angle Pendulums
!=
✓=
r
s
l
g
l
T = 2⇡
r
1
g
f =
2⇡
l
l
g
s = arclength, l = length
Damped Oscillators
t=
c
c=f
!
=
k
p
v/v0
Transverse velocity (movement perpendicular
to wave propagation): vmax = !A = 2A⇡
⇣ = 1 ! critically damped
⇣ < 1 ! underdamped
ẍ + 2⇣!0 ẋ + !02 x = 0
z̈ + aż + b = 0
p
a
!0 = b, ⇣ = p
2 b
A(t) = A0 e !0 ⇣t
p
!damped = !0 1 ⇣ 2
Elastic cord phase velocity:
r
kx(L0 + x)
c=
m
m
T = kx
⇢=
L0 + x
1
Q-factor: Q =
2⇣
High Q-factor ! oscillate for long time
sin (A ± B) = sin A cos B ± cos A sin B
Pstring = Ty v = T !kA2 sin (kx !t + )
s
s
T
B
Phase velocity: c =
=
⇢
⇢
cos (A ± B) = cos A cos B ⌥ sin A sin B
⇣
⌘
⇣
⌘
cos A + cos B = 2 cos A+B
cos A 2 B
2
c
@2
1 @2
= 2
2
@x
c @t2
p
Inst. power (string): P = T ⇢
2. Post-Midterm
Wave Equation:
Wave Phenomena
Mean Intensity:
b
b
⇣= p
=
2m!0
2 mk
⇣ > 1 ! overdamped
f and close A. !
¯
Miscellaneous
2⇡
Apressure = BkA = c⇢!A
✓
f2 | = 2x envelope wave
Waves
lower B ! greater compression. Usually (+)
l
!2
(f ) remains constant, A oscillates at fbeat .
Bulk Modulus: B =
⇣g⌘
! = !1
Beats occur with small
(decays over time)
± (x, t) = A cos (kx ⌥ !t + )
p
! = k T /µ, k = angular wave number
✓¨ =
(!1 + !2 )
,
2
(t) = 2A cos (¯
! t) cos (0.5 !t)
General damped oscillator ! Transient
k=
s
2n⇡
n
=
kd
d
the separation between the sources.
1
2⇣ 2 , resonance if ⇣ < p
2
Resonance ! Steady-state solution
!r = !0
must be a
For destructive, odd:
(n + 12 )
sin ✓ =
, n = 0, 1, 2, ..., where d is
d
Resonance
x(t) = A cos (!t + )
v(t) =
kx
L = d sin ✓, where k is the angular wave
Boundary conditions:
p
Imean = 0.5B!kA2 = 0.5 B⇢! 2 A2
Clamped ! d = 0 !
P
P
I=
(3D) =
(2D)
4⇡r2
2⇡r
1
I / A2 / ! 2 / P / 2 (3D)
r
✓ ◆
I
Decibel Scale: = (10dB) log
I0
W
where I0 = 10 12 2 for sound.
m
@y
Free !
=0!
@x
General form: y(x, t) = 2A sin (kx) sin (!t)
Open end ! anti-node (max A)
!t)
1
Closed end ! node (zero A)
2l
nc
,f =
n
2l
4l
(2n 1)c
=
,f =
, where
2n 1
4l
Both ends open/closed:
Max cancellation when /2 apart.
Phase change due to path length:
= 0 ! no inversion
Standing waves: Nodes separated by /2
Interference (+/- for constr. or destr.):
(x0 , t) = (A1 ± A2 ) cos (kx0
= ⇡ ! inverted
1 open:
=k L
n = 1, 2, ...
=
L
Mirrors: Image opposite object ! virtual
L
Virtual image ! negative v
Doppler E↵ect: f0 =
c0
=
0
✓
c + v0
c vs
◆
fs
the medium, v0 is the velocity of the observer
relative to the medium, and vs is positive for
motion towards the source.
✓
◆
vwave
Shockwave: ✓open = sin 1
vsource
Reflection: ✓i = ✓r
sin ✓1
n2
=
, therefore:
sin ✓2
n1
n2 < n1 ! ✓R > ✓ i , n2 > n1 ! ✓R < ✓ i
Total Internal Reflection: sin ✓crit
Spherical mirrors: r ⇡ 2f
Convex (diverging) mirror ! negative f
Reflection where n2 > n1 !
Lenses: Diverging rays ! virtual
Thin films:
Optical Instruments
n2
=
n1
1
1
1
Mirror/Lens equation:
+ =
u
v
f
I
v
Magnification: M =
=
O
u
n = 1, 2, 3, ..., convert
Lensmaker’s Equation:
✓
◆
1
1
1
= (n 1)
+
, where n is the
f
R1
R2
Wavelength in a medium:
=⇡
concave.
n = 0, 1, 2, 3, ...
Magnifying Glass: Mmax
= /n2
2
n
2 tan ✓
(2n + 1)
Thin wedge constructive: x =
4 tan ✓
d
=1+ ,
f
Thin wedge fringe width:
d
= , d = distance to object.
f
x=
2 tan ✓
p
n r
Newton’s rings, Destructive: x =
q
Constructive: x =
n + 12 r
L
Microscope: M = mo me , where mo =
fo
✓
◆
d
and me = 1 +
, L = lens separation.
fe
fo
do
Telescope: M✓ =
=
(=
if the
✓
fe
de
Central fringe is dark. n = 0 ! first ring
Interferometry: h =
4(n
1)
, where n is the
refractive index. Inversion of pattern when
eyepiece is big enough to collect all light)
arm shifted
1.22
x
=
dlens
r
Angular Resolution: sin ✓R
0
to
Thin wedge destructive: x =
refractive index, R is (+) if convex, (-) if
Mmin
Geometric Optics
(higher n ! less bending)
= (n + 0.5) ! destructive
Diverging lens ! negative f
where c is the velocity of the wave relative to
Snell’s Law:
= n ! constructive
(2m + 1)
, where m = 0, 1, 2, ...
4
Di↵raction
Light Waves
Refractive Index: n =
c
vsource
=
vp
vwave
Single slit:
sin ( ⇡ a sin ✓)
(✓) = A ⇡
cos (kr
a sin ✓
!t)
where a is the width of the slit.
sin2 ( ⇡ a sin ✓)
I = I0
(USE RADIANS!)
⇡2 2
2
2 a sin ✓
n
Min. intensity: sin ✓ =
, n 6= 0
a
Red light (longest ) is least refracted.
d!
dk
!
Phase Velocity: vp =
k
Double slit:
Group Velocity: vg =
(✓) = 2A cos
⇣
⇡d
⌘
sin ✓ cos (kr
!t)
! = vp k
where d is the separation between the centers
vg < vp ! normal dispersion
of two slits.
✓
◆
⇡d
I = I0 cos2
sin ✓ (USE RADIANS!)
vg > vp ! anomalous dispersion
vg = vp ! no dispersion
Polarization: I = I0 cos2
Brewster’s Angle: tan ✓B =
n2
, the reflected
n1
ray will be completely polarized.
Phase di↵erence: Nx =
Lnx
, N in units of
= (N )(2⇡). Phase di↵erence of ⇡ means
N2
N1 =
L
(n2
L. New phase di↵erence:
n1 )
2
zero
Fringe spacing: y /
/ 1/d / 1/a
Di↵raction grating, bright lines:
, L = path length, n = refractive index.
path di↵erence,
n
, n can be zero
d
(n + 0.5)
Min. intensity: sin ✓ =
, n can be
d
Max. intensity: sin ✓ =
sin ✓ =
2
m
= mN , m = 0, 1, 2, ... where N
d
is the slits per unit length. sin ✓ ⇧ 1.
Resolving Power: R =
= mN w, where
m = 0, 1, 2, ... and w is the slit width