Question

Learning Goal: To understand the concept of moment of inertia and how it depends on mass, radius, and mass distribution.

In rigid-body rotational dynamics, the role analogous to the mass of a body (when one is considering translational motion) is played by the body's moment of inertia. For this reason, conceptual understanding of the motion of a rigid body requires some understanding of moments of inertia. This problem should help you develop such an understanding.
The moment of inertia of a body about some specified axis is I = cmr^2, where c is a dimensionless constant, m is the mass of the body, and r is the perpendicular distance from the axis of rotation. Therefore, if you have two similarly shaped objects of the same size but with one twice as massive as the other, the more massive object should have a moment of inertia twice that of the less massive one. Furthermore, if you have two similarly shaped objects of the same mass, but one has twice the size of the other, the larger object should have a moment of inertia that is four times that of the smaller one.
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?

Answer 1

m₂ = 4 kg

Step-by-step explanation:

The moment of inertia is defined by

I = ∫ r² dm

for bodies with high symmetry it is tabulated, for a spherical shell

I = 2/3 m r²

in this case the first sphere has a radius of r₁ = 2m and a mass of m₁ = 1 kg, the second sphere has a radius r₂ = 1m.

They ask what is the masses of the second spherical shell so that the moment of inertia of the two is the same.

I₁ = ⅔ m₁ r₁²

I₂ = ⅔ m₂ r₂²

They ask that the two moments have been equal

I₁ = I₂

⅔ m₁ r₁² = ⅔ m₂ r₂²

m₂ = (r₁ / r₂) ² m₁

let's calculate

m₂ = (2/1) ² 1

m₂ = 4 kg