Question

Two spherical shells have their mass uniformly distrubuted over the spherical surface. One of the shells has a diameter of 2 meters and a mass of 1 kilogram. The other shell has a diameter of 1 meter. What must the mass mmm of the 1-meter shell be for both shells to have the same moment of inertia about their centers of mass

Answer 1

Answer: 4 kg

Step-by-step explanation:

Given

Mass of the first shell, m1 = 1 kg

Diameter of the first shell, d1 = 2 m

Radius of the first shell, r1 = 1 m

Diameter of the second shell, d2 = 1 m

Radius of the second shell, r2 = 1/2 m

The moment of inertia of a spherical shell is given by the relation

I = mr²

This means that if two sphere's have the same moment of ineria:

I1 would be equal to I2. And thus

m1.r1² = m2.r2²

If we solve for the second mass m2

m2 = m1.r1²/r2²

m2 = m1 (r1 / r2)² and we substitute the values

m2 = 1 * (1 / 0.5)²

m2 = 2²

m2 = 4 kg

The needed mass of the second shell for their shells to have the same moment of inertia is 4 kg