⃗ =
⃗ = m a ⃗ = d p⃗
F net
F ext
∑
dt
Newton II (polar/cylindrical)
·
Fr = m( ··r − rϕ2)
∑
··
·
Fϕ = m(rϕ + 2r· ϕ)
∑
Fz = m ··z
∑
p⃗ ≡ m v ⃗
ΔP ⃗ =
∑∫
i
t
t0
∑
Newton II (intrinsic)
d | v⃗ |
F = mat = m
∑ t
dt
v2
F = man = m
∑ n
r
F =0
∑ b
⃗ dt ≡ J ⃗
F ext
net (I.M.T.)
vB2n − vA2n
e=
vA1n − vB1n
M acm
⃗ =
Phys 244 Final Exam Formula Sheet
⃗
miai⃗ = F ext,sys
l ⃗ ≡ r ⃗ × p⃗
τ⃗ ≡ r ⃗ × F ⃗
dl ⃗
τnet
⃗ =
dt
[1 + (dy/dx)2]3/2
r(x) =
| d 2y/dx 2 |
⃗ (v)⃗ = − f (v)v̂
f res
⃗ = q v⃗ × B ⃗
f mag
⃗ =
LO
∑
l i⃗ =
r ⃗ × pi⃗
∑ i
⃗
dLO
(O inertial
τext,O
⃗
=
fixed pt)
∑
dt
⃗
d L cm
τ⃗
=
∑ ext,cm
dt
 ≡
A⃗
| A ⃗|
i ̂ × j ̂ = k̂
j ̂ × k̂ = i ̂
k̂ × i ̂ = j ̂
a ⃗ × b⃗ = − b⃗ × a ⃗
Rigid Body 2D rot
about symmetry axis
Lz =
miri2 ωz = Iωz
(∑
)
i
∑
τext,O = Iα
(O inertial
fixed pt)
τ
= Iαcm
∑ ext,cm
Common Moments of Inertia (2D rot, uniform bodies)
i
2
2
2
1
hollow
I
=
mr
2 thin
2
I
=
r
dm
pt
cts
MR rod: Icm =
ML
Fext,sysdt = dPsys = mdv + vex dm
sphere: Icm =
∫
3
12
2
ring: Icm = MR
parallel axis theorem
solid
2
1
2 thin
2
MR rod: Iend = ML
sphere: Icm =
1
2
2
I
=
I
+
Md
P
cm
disk/cylinder: Icm =
5
MR
3
2
W = F ⃗ ⋅ d r ⃗ = | F ⃗| cos ϕ F &d
⃗ r ⃗ ds = Ft ds
1 2
∫
∫
∫
(W.E.T.) Wtotal = ΔT
Ttrans = mvcm
1 2
1
2
2
⃗ = − ∇U
F cons
T
=
mv
+
I
ω
∂
∂
∂
general
E ≡T+U
1 2
2 cm 2 cm
∇ ≡ i ̂ + j ̂ + k̂
Trot = Iω
∂x
∂y
∂z
Wcons = − ΔU
2
Wnc = ΔE
⃗
F central
( r ⃗ ) = f (r, ϕ, θ)r ̂
∂F
∂F
y
x
⃗
F pos−dep,cons
⇔ ∇ × F ⃗ = 0 (3D) ,
=
(2D)
Wnc = 0 ⇒ E1 = E2 ⇔ ΔT = − ΔU
⃗
F sph
∂y
∂x
sym( r ⃗ ) = f (r)r ̂
Energy cont.
Uint,system =
∑∑
i
j>i
Uij
curl and grad in
spherical polars
1
E=
mivi2 +
U +
U
∑2
∑ ∑ ij ∑ i,ext
i
i j>i
i
Oscillations
Pure SHM
∇U =
x(t) = C1e
2
mω = k
iωt
+ C2e
Damped SHM
β 2 − ω02 )t
[
x2
∫x
S[y] =
1
1
Lagrangian Mechanics
S≡
t2
∫t
L(qi, q· i, t) dt
L ≡T−U
1
ext
x·· + 2β x· + ω02 x = f (t)
]
δS = δ
t2
∫t
Modified E-L
qi → q′i = qi + εKi(q)⃗
∂L
Qcons =
K
∑ ∂q· i i
i
d
∂L
∂L
∂f
=
+
λ(t)
dt ( ∂q· i ) ∂qi
∂qi
d
∂L
∂L
=
dt ( ∂q· i ) ∂qi
Ldt = 0 ⇒
f (q1, q2) = c



𝕄

𝕂





𝕄
𝕂

⃗ = (m r ⃗ + m r ⃗ )/M
R ⃗ = R CM
1 1
2 2
A(ω) ≡
f0
(ω02 − ω 2)2 + 4β 2 ω 2
tan δ = 2βω /(ω02 − ω 2 )
Hamiltonian Mechanics
pi ≡ ∂L /∂q· i
H(q,⃗ p,⃗ t) ≡
·
piq·i − L(q,⃗ q,⃗ t)
(∑
)
i
Mechanical Waves (Idealized)
· 2 1
1
· 2
⃗
L = M R + μ r ⃗ − U(r)
2
2
r ⃗ ≡ r 1⃗ − r 2⃗
x(t) = At e −βt cos(ω1t + δt ) + A(ω)cos(ωt − δ)
∂H
∂H
q·i =
, p· i = −
∂pi
∂qi
Two-Body Central Force Problem
m1m2
m1 + m2
( − ω2 ) ⋅ a⃗ = 0
f (t) ≡ Fext (t)/m
f (t) = f0 cos(ωt)
Euler-Lagrange Equations
1
Noether
μ≡
⇒
ai⃗ e iωit
F(y, y′, x) = F(y′, x) ⇒
d
∂F
∂F
=
d x ( ∂y′ ) ∂y
1
Forced Damped SHM
m x·· = − k x − b x· + F (t)
⋅ x⃗ ⇒
Gm1m2
kq q
, Ues = 1 2
r
r
∂F
= const
∂y′
Special Cases w/ First Integrals
∂F
F(y, y′, x) = F(y, y′) ⇒ y′
− F = const
∂y′
Euler Equation
F(y, y′, x) d x
··
⋅ x⃗ = −
ω02 − β 2
ω1 ≡
Std Potentials: Ug = −
Coupled Oscillators
β 2 − ω02 )t
−( β−
x(t) = C1e
+ C2e
m x·· + bx· + k x = 0 ⇒
x(t) = e βt [C + Dt]
2
x·· + 2βx· + ω0 x = 0
x(t) = e −βt A cos(ω t) + B sin(ω t)
Calculus of Variations
∂U
1 ∂U ̂
1 ∂U
r̂ +
θ+
∂r
r ∂θ
r sin θ ∂ϕ
−iωt
−( β+
b
k
≡ 2β , ω02 =
m
m
∂F
∂F
1
∂
1 ∂Fr 1 ∂
1 ∂
(sin θ ⋅ Fϕ) − θ r ̂ +
−
(rFϕ) θ ̂ +
(rFθ ) − r
[ r sin θ ∂ϕ
]
r sin θ [ ∂θ
∂ϕ ]
r ∂r
r [ ∂r
∂θ ]
F ⃗ ≡ − kr ⃗
x(t) = A cos(ωt − δ)
x(t) = B1 cos(ωt) + B2 sin(ωt)
m x·· = − k x ⇒
∇× F ⃗=
Ueff (r) = U(r) +
lz
γμ
,ε =
Alz
γμ
F/μ , v = λ f
ω = 2π /T = 2π f , k = 2π /λ
2μr 2
γ
c
U(r) = −
⇒ r(ϕ) =
r 2
1 + ε cos ϕ
2
c=
v=
lz2
E=
2
γ μ 2
(ε − 1)
2lz2
y(x, t) = A sin(k x ± ωt)
y(x, t) = ASW sin(k x)sin(ωt)
vp = ω/k , vg = dω/dk = vp + kdvp /dk