| Translational and Rotational Quantities Compared | ||||||
| translational | connections | rotational | ||||
|---|---|---|---|---|---|---|
| base quantities | s, r |
s = |
θ × r |
θ |
||
| coordinate systems | r = |
x ˆi + y ˆj |
x = r cos θ y = r sin θ r = √(x2 + y2) θ = tan−1 (y / x) |
r = |
rˆr + θˆθ |
|
| velocity | v = |
dr / dt |
v = |
ω × r |
ω = |
dθ / dt |
| acceleration | a = |
dv / dt = d2r / dt2 |
a = |
α × r − ω2r |
α = |
dω / dt = d2θ / dt2 |
| equations of motion | v = v0 + at x = x0 + v0t + ½ at2 v2 = v02 + 2a(x − x0) |
ω = ω0 + αt θ = θ0 + ω0t + ½ αt2 ω2 = ω02 + 2α(θ − θ0) |
||||
| cause of acceleration | ∑ F |
τ = |
r × F |
∑ τ |
||
| resistance to acceleration | m |
I = |
∑ ri2mi = ∫ r2 dm |
I |
||
| newton's second law | ∑ F = |
m a |
∑ τ = |
I α |
||
| equilibrium | ∑ F = 0 ⇒ |
⎧∑ F+x = ∑ F−x ⎨∑ F+y = ∑ F−y ⎩∑ F+z = ∑ F−z |
∑ τ = 0 ⇒ |
⎧∑ τ+x = ∑ τ−x ⎨∑ τ+y = ∑ τ−y ⎩∑ τ+z = ∑ τ−z |
||
| momentum | p = |
m v |
L = |
r × p = m r × v |
L = |
I ω |
| impulse-momentum | ∫ F · dt F = |
= Δp dp / dt |
∫ τ · dt τ |
= ΔL dL / dt |
||
| work-energy | W = |
∫ F · ds |
W = |
∫ τ · dθ |
||
| kinetic energy | K = |
½ mv2 |
K = |
½ Iω2 |
||
| potential energy | U = F(x) = |
− ∫ F · ds − dU / dx |
U = τ(θ) = |
− ∫ τ · dθ − dU / dθ |
||
| power | P = |
F · v |
P = |
τ · ω | ||