Question

A spherical shell of radius 2.59 cm and a sphere of radius 9.22 cm are rolling without slipping along the same floor. The two objects have the same mass. If they are to have the same total kinetic energy, what should the ratio of the spherical shell\'s angular speed to the sphere\'s angular speed be?

Answer 1

Angular kinetic energy can be written as:
KE = 1/2 I ω²,
where KE is the kinetic energy,
I is the moment of inertia (depends on geometry and mass),
and ω is the angular speed.

The moments of inertia are:
Spherical shell: I = 2/3 mr²
Cylinder (solid): I = 1/2 mr²

If they are to have the same kinetic energy:
KE(sphere) = KE(cylinder)
1/2 I(s) (ω(s))² = 1/2 I(c) (ω(c))²
1/2 ( 2/3 m(s) (r(s))² ) (ω(s))² = 1/2 ( 1/2 m(c) (r(c))²) (ω(c))²

Since m(s) = m(c):
2/3 (r(s))² (ω(s))² = 1/2 (r(c))² (ω(c))²
(ω(s)/ω(c))² = 3/4 (r(c)/r(s))²
ω(s)/ω(c) = √3/2 r(c)/r(s)

r(s) = 2.59 cm, and r(c) = 9.22 cm:
ω(s)/ω(c) = √3/2 (9.22/2.59)
ω(s)/ω(c) = 3.08