Rotational Energy
Discussion
Rotational kinetic energy
For a system of point bodies
| K = | 1 | ∑mivi2 |
| 2 |
| K = | 1 | ∑ri2mi | vi2 |
| 2 | ri2 |
| K = | 1 | Iω2 |
| 2 |
For an extended body
| K = | 1 | ⌠ ⌡ |
v2 dm |
| 2 |
| K = | 1 | ⌠ ⌡ |
v2 | r2 dm |
| 2 | r2 |
| K = | 1 | ω2I |
| 2 |
| K = | 1 | Iω2 |
| 2 |
Moment of inertia
| I = | ⌠ ⌡ |
r2 dm = ρ | ⌠ ⌡ |
r2 dV |
Angular work
| W = | ⌠ ⌡ |
F ⋅ ds |
| W = | ⌠ ⌡ |
F ⋅ (dθ × r) |
| W = | ⌠ ⌡ |
dθ ⋅ (r × F) |
| W = | ⌠ ⌡ |
dθ ⋅ τ |
| W = | ⌠ ⌡ |
τ ⋅ dθ |
and so on.
| concept | translation | connection | rotation | |||
|---|---|---|---|---|---|---|
| work-energy | W = | ∫ F ⋅ ds | W = | ∫ τ ⋅ dθ | ||
| kinetic energy | K = | ½mv2 | K = | ½Iω2 | ||
| potential energy | U = F(x) = |
− ∫ F ⋅ ds − dU/dx |
U = τ(θ) = |
− ∫ τ ⋅ dθ − dU/dθ |
||
| power | P = | F ⋅ v | P = | τ ⋅ ω | ||