PY 205 Sample Equation sheet Fall 2022
Things you must know:
Definition of average velocity
Definition of particle energy
The Energy Principle
Definition of momentum
Definition of kinetic energy
The Momentum Principle
Definition of work
Other fundamental principles:
⃗A
dL
= ⃗τnet,A
dt
⃗ A = ⃗τnet,A ∆t
∆L
⃗τA = ⃗rA × F⃗
⃗ trans,A = ⃗rA × p⃗ (point particle)
L
Multiparticle systems:
⃗rcm =
m1⃗r1 + m2⃗r2 + . . .
m1 + m2 + . . .
Krel = Krot + Kvib
⃗ trans,A = ⃗rcm,A × P⃗tot
L
p⃗tot ≈ Mtot⃗vcm (|⃗v | ≪ c)
Ktot = Ktrans + Krel
1
Ktrans ≈ Mtot |⃗vcm |2 (|⃗v | ≪ c)
2
2 + m r2 + . . .
⃗ rot = I⃗
L
ω
I = m1 r1⊥
2 2⊥
L2
1
Krot = rot = Iω 2
2I
2
⃗A = L
⃗ trans,A + L
⃗ rot
L
R
R
L
R
R
Sphere
2
I = M R2
5
Cylinder/Disc
1
I = M R2
2
Rod/Cylinder/Disc
1
1
I = M L2 + M R 2
12
4
1
2
I = M L for R ≪ L
12
Thin Hoop
I = M R2
Other physical relationships:
x(t) = xi + vix t +
γ≡s
1−
1
|⃗v |
c
1 Fnet,x
2
2 m (t)
2
m1 m2
F⃗grav = −G
r̂
|⃗r|2
m1 m2
Ugrav = −G
|⃗r|
⃗ − L0
F⃗spring = −ks sL̂, s = L
sliding Ffriction ≈ µk FN
Y =
F/A
(macro)
∆L/L
→
−
−
→
pi
→
−
−
rf =→
ri+
∆t + 12 Fmnet (∆t)2
m
2
d⃗v
2
2
E − (pc) = mc2
⃗a =
dt
d⃗
p
p|⃗v |
m|⃗v |2
=
≈
(if |⃗v | ≪ c) where R = radius of kissing circle
dt ⊥
R
R
F⃗grav ≈ mg near Earth
∆Ugrav ≈ mg∆y near Earth
1
Uspring = ks s2
2
static Ffriction ≤ µs FN
Y =
ksi
(micro)
d
1
1 q1 q2
r̂
4πϵ0 |⃗r|2
1 q1 q2
Uelec =
r 4πϵ0 |⃗r|
2π
1
ks
ω=
with ω =
and f =
m
T
T
F⃗air = 12 CρA |⃗v |2
r
ksi
speed of sound v = d
ma
F⃗elec =
1
Ui ≈ ksi s2 − EM interatomic pot. energy
2
13.6eV
where N = 1, 2, 3 . . . (Hydrogen atom energy levels)
EN = −
N2
r
ksi
EN = N ℏω0 + E0 where N = 0, 1, 2 . . . and ω0 =
(Quantized oscillator energy levels)
ma
Ω=
(q + N − 1)!
q! (N − 1)!
∆Ethermal = mC∆T
1
∂S
≡
T
∂E
S ≡ k ln Ω
E
prob(E) ∝ Ω (E) e kB T
Constant
Speed of light
Gravitational constant
Approx. grav field near Earth’s surface
Electron mass
Proton mass
Neutron mass
Proton charge
Electron volt
Avogadro’s number
Plank’s constant
h
hbar =
2π
specific heat capacity of water
Boltzmann constant
Room temperature
milli
micro
nano
pico
m
µ
n
p
1 × 10−3
1 × 10−6
1 × 10−9
1 × 10−12
kilo
mega
giga
tera
Symbol
c
G
g
me
mp
mn
1
4πϵ0
e
1 eV
NA
h
Approximate Value
3 × 108 m/s
6.7 × 10−11 N · m2 /kg2
9.8 N/kg
9.1 × 10−31 kg
1.67 × 10−27 kg
1.67 × 10−27 kg
9 × 109 N · m2 /C2
1.6 × 10−19 C
1.6 × 10−19 J
6.02 × 1023 atoms/mol
6.6 × 10−34 J · s
ℏ
1.05 × 10−34 J · s
C
kB
4.2 kJ/kg/◦ C
1.38 × 10−23 J/K
293.15 K
Troom
K
M
G
T
Q
(small Q)
T
−
∂E
C=
/atom
∂T
Electric constant
∆S =
1 × 103
1 × 106
1 × 109
1 × 1012
fˆ = ⟨cos θx , cos θy , cos θz ⟩ unit vector from angles
⃗·B
⃗ = Ax Bx + Ay By + Az Bz = A
⃗ B
⃗ cos θ
A
⃗ B
⃗ = ⟨Ay Bz − Az By , Az Bx − Ax Bz , Ax By − Ay Bx ⟩
A×
⃗×B
⃗ = A
⃗ B
⃗ sin θ
A
2