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