1. Oscillatory Motion
F
k
x
v
a
t
A, r
ω
φ
T
m
L
vt
at
ar
α
Ek
Ep
I
d
Ff
b
ω
e
Fd
F◦
ω◦
ωd
F = −kx
x(t) = A sin(ωt − φ)
v(t) = Aω cos(ωt − φ)
a(t) = −Aω 2 sin(ωt − φ)
vt = ωr
at = rα
ar = ω 2 r
1
Ek = Iω 2
2
1
Ek = kA2 sin2 (ωt)
2
1 2
Ep = kA cos2 (ωt)
2
Ff = −bv
−bt /
2m
x(t) = Ae
r
cos(e
ω t + φ)
k
b2
−
m 4m2
Fd (t) = F◦ cos(ωt)
F◦
A(ωd ) = r
2 2
d
m ωd2 − ω◦2 + bω
m
ω
e=
r
r
r
r
k
g
mgd
κ
2π
ω=
=
=
=
=
m
L
I
I
T
s
s
L
I
T = 2π
= 2π
g
mgd
1
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Force, N
Elastic modulus
Displacement, m
Velocity, ms−1
Acceleration, ms−2
Time, s
Amplitude (radius) of oscillation, m
Angular frequency, s−1
Phase shift, rads
Period, s
Mass, kg
Length of pendulum, m
Tangential velocity, ms−1
Tangential acceleration, ms−2
Radial acceleration, ms−2
Angular acceleration, ms−2
Kinetic energy, J
Potential energy, J
Moment of inertia
Distance from pivot to center of mass, m
Damping force, N
Constant of proportionality, kgs−1
Varying angular frequency of system, s−1
Driving force, N
Initial force, N
Natural frequency
Driving frequency
2. Waves
y
f (x)
κ
vw
λ
Ft
µ
P
Ek
y(x, t) = f (x − vw t)
y(x, t) = A cos(κx − ωt − φ)
s
Ft
λ
ω
vw = = =
T
κ
µ
1
P = µω 2 A2 vw
2
dEk = lim ∆Ek
∆x→0
Z λ
Ek (λ) =
dEk
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Displacement of medium from equilibrium, m
Function of wave pulse at any point in time
Wave number
Wave speed, ms−1
Wavelength, m
Force of tension, N
Linear density, kgm−1
Power derived from wave, Js−1
Kinetic energy of the wave, J
0
A2 ω 2 µπ
=
2k
3. Sound waves
s(x, t) = smax cos(κx − ωt)
−V ∆P
B=
s∆V
γP
ρ
vw =
r
ϡkB T
m
∆P(x, t) = ∆Pmax sin(κx − ωt)
∆Pmax = ρvs ωsmax
1
P = ρAvw (ωsmax )2
2
P
P
=
I=
A 4πr2 vo
·moving source
f0 = f
·no change in λ
vo ± vs
vo ± vs
·moving observer
f0 = f
·change in λ
vo
vw =
2
s
B
V
P
γ
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ρ
ϡ
A
I
f
f0
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vs
vo
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Displacement from equilibrium, m
Bulk modulus, P a
Volume, m3
Sound pressure, P a
Gas characteristic constant
Monatomic gases : 5 /3
Diatomic gases : 7 /5
Density of medium, kgm−3
Adiabatic index (isentropic expansion factor)
Cross-sectional area of path of wave, m2
Wave intensity, Js−1 m−2
Frequency, s−1
Doppler effect frequency, s−1
+ : Source receding / observer approaching
− : Source approaching / observer receding
Speed of source, ms−1
Speed of observer, ms−1
4. Superposition
superposition If two or more travelling waves are moving through a medium
principle :
the resulting wave function is the sum of two individual wave
functions. Holds for small oscillations.
dispersion :
The phenomenon that wave speed depends on frequency.
1
y(x, t) = 2A sin(κx + φ̄) sin(ωt + ∆φ12 )
2
φ1 + φ2
φ̄ =
2
∆φ12 = φ2 − φ1
mλ
L=
2
∆φ = κ∆L
y
φ1 , φ2
L
m
∆φ
∆L
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Equation of motion for interfering waves
Phase shifts of two interfering waves
Length of string, m
Mode number, m ∈ N
Phase difference of two interfering waves
Path difference of two interfering waves
5. Interference
light :
In a vacuum, consists of oscillating electric and magnetic fields, such that the
directions of the two different oscillations are perpendicular to each other.
Huygens’
Each point on a wavefront acts as a disturbance of the medium in which it is
principle :
propagating and thus is a source for further waves.
diffraction :
Ability of waves to travel around corners.
monochromatic : Of a single wavelength. White light is polychromatic.
coherent :
Having a constant phase relationship.
** Number of fringes corresponds to number of wavelengths by which two paths differ.
λ
vw
θ
vw1
sin(θ1 )
λ1
=
=
λ2
vw2
sin(θ2 )
c
ni =
vwi
n1 sin(θ1 ) = n2 sin(θ2 )
∆φ = κn`
ni
c
vwi
∆φ
`
∆L
d
`OP L
∆L = d sin(θ) = mλ , m ∈ N
`OP L = n`
3
: Wavelength of interfering waves
: Wave speed of interfering waves
: Angle from normal of incident
and refracted waves
: Index of refraction of material i
: Speed of light in a vacuum, ms−1
: Speed of wave in material i, ms−1
: Phase shift of wave
: Length traveled by wave in medium, m
: Path difference of wave
: Distance between two slits, m
: Optical path length
6. Quantum Mechanics
h
qe
Vs
Φ
f◦
p
∆x
EB
Ry
n
kB
T
ψ(x)
ψ(x)∗
V (x)
hai
â
E = hf = ~ω
h
~=
2π
qe Vs = hf − Φ
Φ = hf◦
h
p=
λ
~
6 ∆x∆p
2
6 ∆E∆t
p2
2m
Ry
EB = − 2
n
2
mke qe2
En = −
2
n~
· Electron-proton energies
p
E 2 − (pc)2
m=
c2
· Invariant mass of a system
E=
Planck constant, Js
Charge of an electron C
Stopping voltage, V
Work function, J
Critical frequency, Hz
Momentum, kgms−1
Uncertainty in observed value x
Binding energy, J
Rydberg constant, eV
Energy level, n ∈ N
Boltzmann constant
Temperature, K
Wave function
Complex conjugate of wave function
Potential energy
Expectation value of observable a
Operator of observable, matrix form
Observable Operator
x
hxi
x2
x2
δ
hpi
δx (−i)~
h(1 − cos θ)
me c
· Compton scattering
∆λ =
· Square wells
Infinite : ψ 00 (x) = −k 2 ψ(x)
2mE
k2 =
~2
−2
λmax T = 0.2898 · 10 mK
· Wien’s Displacement Law
2πckB T
I(λ, T ) =
λ4
· Rayleigh-Jeans Law
Finite : ψ 00 (x) = α2 ψ(x)
α2 =
2πhc2
I(λ, T ) =
λ5 (ehc/λkB T − 1)
· Planck’s Law
Eψ(x) =
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−~2
2m(V − E)
~2
· Special Relativity
Lorentz factor :
δ2
ψ(x) + V (x) + ψ(x)
2m δx2
· Schrödinger equation
Z ∞
1=
|ψ(x)|2 dx
time
dilation
1
γ=q
1−
v2
c2
∆t◦ proper time
∆t = ∆t◦ γ
∆t observed time
mass
m◦ proper mass
m = m◦ γ
expansion
m observed mass
L◦
length
L◦ proper length
L=
contraction
L observed length
γ
−∞
· Normalization condition
Z ∞
hai =
ψ ∗ (x)âψ(x) dx
−∞
4
7. Universal Gravitation
GM m
r2
−GM m
=
r G
Epg
r
T
a
vc
ve
L
F =
Epg
1
1
−
r1 r2
∆Epg = GM m
2
4π
2
a3
T =
GM
r
GM
vc =
r
r
2GM
ve =
r
L = mvr
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Universal gravitational constant
Gravitational potential energy
Distance between center of mass of two objects
Period of revolution, s
Length of semi-major axis, m
Speed in a circular orbit, ms−1
Escape speed, ms−1
Angular momentum, kgm2 s−1
8. Fluid Mechanics
p
p◦
ρ
h
A
FB
Vm
η
r
C
τ
δv /
δx
Re
d
R
∆p = ρg∆h
p = p◦ + ρgh =
F
A
FB = Vm ρg
1
const = p + ρv 2 + ρgh
2
Fdrag = 6πηrv
· laminar flow
1
= CAρv 2 · turbulent flow
2
F
τ=
A
τ
η = δv
/δx
ρvd
Re =
η
R = A1 v1 = A2 v2
5
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Pressure, kgm−1 s−2
Pressure at surface of liquid
Density of fluid, kgm−3
Change in height, m
Area, m2
Buoyancy force, N
Volume of mass in water
Viscosity, kgm−1 s−1
Radius of object experiencing drag force
Drag coefficient
Shear stress
Velocity gradient
Reynolds number
Characteristic length, m
Flow rate, m3 s−1
9. Constants
h
c
me
mp
qe
qp
ke
µ0
ε0
RE
mE
g
G
Ry
h = 6.62606896 · 10−34 J · s
= 4.13566733 · 10−15 eV · s
c = 2.99792458 · 108 m · s−1
me = 9.10938215 · 10−31 kg
= 0.510998910 M eV · c−2
mp = 1.672621638 · 10−27 kg
= 938.272013 M eV · c−2
qe = −1.602176487 · 10−19 C
qp = 1.602176487 · 10−19 C
ke = c2 · 10−7
= 8.9875517873 · 109 N · m2 · C −2
µ0 = 4π · 10−7
ε0 = (µ0 · c2 )−1
= 2.2654418729 · 10−3
RE = 6.371 · 106 m
mE = 5.9742 · 1024 kg
g = 9.80665 ms−2
G = 6.67428 · 10−11 m3 · kg −1 s−2
Ry = 13.6056923 eV
6
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Planck constant
Speed of light in a vacuum
Electron mass
Proton mass
Electron charge
Proton charge
Coulomb’s constant
Permeability constant of free space
Permittivity constant of free space
Earth radius
Earth mass
Earth surface gravity
Universal gravitational constant
Rydberg constant