Formula sheet
~a
~rf − ~ri
∆~r
Displacement ~rf − ~ri = ∆~r
Velocity ~v =
=
|~a|
tf − ti
∆t
For motion along a curved path,
d~p p̂ = F~k and |~p| dp̂ = |~p| |~v | n̂ = F~⊥ or p v = F⊥
dt dt
R
R
Unit vector â =
where R is the radius of the “kissing circle” and n̂ points in the direction of F~⊥ (toward
center of circle).
Speed of light c = 3×108 m/s G = 6.7×10−11
N · m2
kg2
N · m2
1
= 9×109
4π0
C2
0◦ C = 273 K
Mm
GM m
Magnitude of gravitational force F~gr = G 2 Gravitational potential energy Ugr = −
|~r|
|~r|
N
Near the Earth’s surface F~gr ≈ mg and ∆Ugr ≈ ∆(mgy)
g = +9.8
kg
Magnitude of electric force F~el =
1 |Qq|
4π0 |~r|2
Electric potential energy Uel =
1 Qq
4π0 |~r|
1
Spring potential energy Uspring = ks s2 + U0
2
Magnitude of spring force F~spring = ks |s|
where a negative U0 makes Uspring < 0. It is the same value for initial and final states, so it
cancels from the energy principle formula.
r
Idealized mass-spring oscillator x = A cos ωt = A cos
!
ks
t ,
m
1
ω
=
T
2π
r
ks,i
=d
m
f=
stress
F/A
ks,i
=
=
(in terms of atomic quantities)
vsound
strain
∆L/L
d
Kinetic energy, valid at any speed less than c K = E − mc2
1
Kinetic energy, valid at speeds much less than c K ≈ m|~v |2
2
2
Relationship between relativistic energy and momentum of a particle E 2 − (pc)2 = (mc2 )
Y =
More equations on next page
1
~ A = h(ypz − zpy ), (zpx − xpz ), (xpy − ypx )i
L
I=
X
~τA = ~rA × F~
L2
1
Krot = Iω 2 = rot
2
2I
mi ri2 = m1 r12 + m2 r22 + · · ·
i
# ways to arrange q quanta of energy among N one-dimensional oscillators
Ω=
(q + N − 1)!
q!(N − 1)!
Entropy S = k ln Ω where k = 1.38 × 10−23 J/K
Temperature :
∆S
1
=
T
∆E
Specific heat capacity per atom : C =
∆Eatom
∆T
Probability of finding energy E in small system in contact with large reservoir is proportional
to Ω(E)e−E/kT
2