Quadratic formula: Kinematics under constant acceleration: Average ...

Angular velocity and acceleration:
⇥ = d dt
= d⇥
dt
Angular ↔ Linear analogies:
Center of Mass, Momentum and Impulse:
1D Collision, elastic:
Angular motion at constant angular acceleration:
⇥(t) =⇥(t 0 )+ [t t 0 ]
⇤(t) 2 = ⇤(t 0 ) 2 +2 [⇥(t) ⇥(t 0 )]
⇥(t) =⇥(t 0 )+⇤(t 0 )[t t 0 ]+ 1 2
[t t 0 ] 2
Since these equations describe rotation
x v a m I
Torque and angular momentum:
⇥F ⇥ p L
⇤⇥ = I⇤ = d⇤ L
dt = ⇤r F ⇤ ⇥
1
L = I ⇥ = ⇥r ⇥p
2 mv2 $ 1 2 I!2 Potential Energy & Energy Conservation: F = dU
dx
Z
R center of mass = 1 m i r i = 1 xf
Z ~r f
M total
M
rdm
U = U f U i = Fdx = F ~ · dr ~ = Wcons
x i ~r i
i
~p = m~v
Z
K 1 + U 1 + W non-cons = K 2 + U 2 U gravity = mgy
t2
~J = ~p 2 ~p 1 = F ~ (t)dt
~F U spring = 1 2 k(x x 0) 2 U gravity =
GmM
net = d~p
t 1
dt
r
Power:
P average =
W P = F · v
v f = m 1v 1i + m 2 v 2i
P = dW m 1 + m 2
t
dt P = ⇤ · ⇤⇥
v 1f = m 1 m 2
v 1i + 2m 2
v
The formula for v2f is obtained by swapping 1↔2 in
2i
m 1 + m 2 m 1 + m 2 the formula to the left.
ring or hollow cylinder (about center): mr 2
2
solid sphere (about center):
I = m i ri 2 = r 2 5
dm
mr2
1
i
disc, solid cylinder (about center): 2 mr2 2
hollow sphere (about center): 3 mr2
I = I CoM + MD 2 1
1
thin rod (about center): 12 mL2 thin rod about end: 3 mL2
1D Collision, totally inelastic:
Moment of inertia:
a tangential = ↵r
v tangential = !r
around a fixed axis, they involve the
components of θ, ω and α along this axis.