A uniform solid sphere of mass M and radius R rotates with an angular speed abou

Question

A uniform solid sphere of mass M and radius R rotates with an angular speed about an axis through its center. A uniform solid cylinder of mass M, radius R, and length 2R rotates through an axis running through the central axis of the cylinder. What must be the angular speed of the cylinder so it will have the same rotational kinetic energy as the sphere? (the moment of inertia of a solid sphere with mass M and radius R is 2/5MR^2 and the moment of inertia of a solid cylinder with mass M and radius R is 1/2 MR^2)

Explanation / Answer

Kinetic energy of rotation is
K = 1/2 Iw2; where I is the moment of inertia and w the angular speed of rotation.

The letter "c" will represent the cylinder, and the letter "s" the sphere:

Kc = Ks

Kc = 1/2 Ic*wc2

Ks = 1/2 Is*ws2

Then:

1/2 Ic*wc2 = 1/2 Is*ws2 Solving for wc we have:

wc = (Is/Ic)1/2 ws

The moment of Innertia for both are:

Is = 2/5 mr2

Ic = 1/2 mr2

wc = (2/5 mr2 / 1/2 mr2)1/2 ws

wc = (2/5 / 1/2)1/2 ws

wc = (4/5)1/2 ws

wc = 2/(5)1/2 ws

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