The Physics
Hypertextbook
Opus in profectus

Rotational Energy

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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.

Translational and rotational quantities compared
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