A spherical shell of radius 3.84 cm and a sphere of radius 7.72 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 spherical shell's angular speed to the sphere's angular speed be?

The ratio of the spherical shell's angular speed to the sphere's angular speed be;

(ω_ss)/(ω_s) = 1.557

We are given;

Radius of spherical shell; r = 3.84 cm = 0.0384 m

Radius of sphere; r = 7.72 cm = 0.0772 m

Formula for rotational kinetic energy is;

KE = ½Iω²

Now, moment of inertia for spherical shell is;

I_ss = ⅔mr²

While moment of inertia for solid sphere is;

I_s = (2/5)mr²

Thus;

I_ss = ⅔m(0.0384²)

I_s = (2/5)m(0.0772²)

We are told that their kinetic energy is equal. Thus;

½(⅔m(0.0384²))(ω_ss)²= ½((2/5)m(0.0772²))(ω_s)²

m will cancel out to give;

⅓(0.0384²)(ω_ss)² = (1/5)(0.0772²)(ω_s)²

0.00049152(ω_ss)² = 0.001191968(ω_s)²

Thus;

(ω_ss)²/(ω_s)² = 0.001191968/0.00049152

(ω_ss)²/(ω_s)² = 2.425065

(ω_ss)/(ω_s) = √2.425065

(ω_ss)/(ω_s) = 1.557

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Explanation:

Let assume that both the spherical shell and the sphere are rigid bodies. The kinetic energy due to rolling is:

The moments of inertial for the spherical shell and the sphere are, respectively:

Spherical Shell

Sphere

Given that both rigid bodies have the same kinetic energy and the same mass, then:

The ratio of the spherical shell's angular speed to the sphere's angular spped is:

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