Rotational Energy

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Rotational Energy

Discussion

Rotational kinetic energy

For a system of point bodies

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

For an extended body

K = 1
v2 dm
2
K = 1
v2r2 dm
2r2
K = 1ω2I
2
K = 1Iω2
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
concepttranslationconnectionrotation
work-energyW = ∫ F · ds  W = ∫ τ · dθ
kinetic energyK = ½mv2  K = ½Iω2
potential energyU =
F(x) = 
− ∫ F · ds
− dU/dx
  U =
τ(θ) = 
− ∫ τ · dθ
− dU/dθ
powerP = F · v  P = τ · ω

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