A sphere of radius 4.34 cm and a spherical shell of radius 6.72 cm are rolling w

Question

A sphere of radius 4.34 cm and a spherical shell of radius 6.72 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 sphere's angular speed to the spherical shell's angular speed be?

Hints:The moments of inertia of a sphere and a spherical shell about their respective centers of mass are:

I of sphere=2/5MR^2,sphere and I of shell = 2/3 MR^2, shell.

Objects undergoing both rotational and translational motion have both rotational and translational kinetic energy. These kinetic energies depend on the mass, speed, moment of inertia, and angular speed of the objects as follows:

Ktrans= 1/2 Mv^2 cm and Krotational =1/2 Icm w^2

The speed to be used is the object's center of mass speed, and the moment of inertia should be for an axis of rotation passing through the object's center of mass. The translational speed of a rolling object's center of mass is related to its angular speed as follows:

vcm= wR

Explanation / Answer

Let 1 refer to sphere and 2 the shell
KE1 = 1/2 M1 V1^2 + 1/2 M1 * 2/5 V1^2 = 7 M1 V1^2 / 10
Note that 1/2 * I1 * w^2 = 1/2 M1 * 2/5 * R1^2 * (V1 / R1)^2 = 1/5 M1 V1^2
KE2 = 1/2 M2 V2^2 + 1/2 * M2 * 2/3 V2^2 = 5/6 M2 V2^2
7 V1^2 / 10 = 5/6 V2^2        since the masses and energies are equal
V1 = (25/21)^1/2 V2 = 1.09 V2
w1 R1 = 1.09 w2 R2    where w is angular speed
w1 / w2 = 1.09 * 4.34 / 6.72 = .704

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