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