Rotational Energy

Discussion

Rotational kinetic energy

Translational and Rotational Quantities Compared
  translational connections rotational
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

For a system of point bodies

K =  1  ∑ mivi2 =  1  ∑ ri2mi  vi2  =  1  Iω2
2 2 ri2 2

For an extended body

K =  1 ⌠
⌡		  v2 dm =  1 ⌠
⌡		 v2  r2 dm =  1  ω2I =  1  Iω2
2 2 r2 2 2

Moment of inertia

I = ∑r2m =  ⌠
⌡		  r2 dm = ρ  ⌠
⌡		  r2 dV

Angular work

W =  ⌠
⌡		  F · ds =  ⌠
⌡		  F · (dθ × r) =  ⌠
⌡		  dθ · (r × F) =  ⌠
⌡		  dθ · τ =  ⌠
⌡		  τ · dθ

and so on

The Physics Hypertextbook

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