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 m of the 1-meter shell be for both shells to have the same moment of inertia about their centers of mass?

Question

Mathematics

Emilie Bauer

28 September, 15:21

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 m of the 1-meter shell be for both shells to have the same moment of inertia about their centers of mass?

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Zaire Bird · Answer 1

The moment of inertia of a spherical shell is I = mr²

If two spheres have the same moment of inertia, then

I₁ = I₂

m₁ (r₁) ² = m₂ (r₂) ²

Solve for the second mass:

m₂ = m₁ (r₁/r₂) ²

Given m₁ = 1 kg, r₁ = 2 m, r₂ = 1m,

m₂ = (1 kg) (2 m / 1 m) ² = 4 kg