For the following objects, which all roll without slip- ping, determine the rotational kinetic energy about the center of mass as a percentage of the total kinetic energy: (a) a solid sphere, (b) a thin spherical shell, and (c) a tiin cylindrical shell.

The answers to the question are

(a) 28.57 %

(b) 40 %

(c) 50 %

Explanation:

The following are the relations between total and rotational kinetic energies of a sphere

=

ω = angular velocity

v = velocity

m = mass

I = Moment of inertia =  and ω = v/r

Therefore =

Therefore

= 28.57 %

(b) for a thin sherical shell, I = Therefore

= = = 1/3mv²

KE(Total) = KE(Transitional) + KE(Rotational) = 1/2mv² + 1/3mv² = 5/6mv²

KE(Transitional) /KE(Total) = (1/3mv²)/(5/6mv²) = 6/15

Percentage total = 6*100/15 = 40 %

(c) Thin cylindrical shell = I = and

Therefore

and

In percentage = = 50 %