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 m of the 1-meter shell be for both shells to have the same moment of inertia about their centers of mass? Part B Consider the moment of inertia of a solid uniform disk, versus that of a solid sphere, about their respective centers of mass. Assume that they both have the same mass and outer radius, that they have uniform mass distributions, and that the disk is rotated about an axis perpendicular to its face. What is the relation between the moment of inertia of the disk I_disk and that of the sphere I_sphere?

Answer 1

a) m₂ = 4 kg , b) I_disk / I sphere = 5/4

Step-by-step explanation:

a) This exercise is requested to make several comparisons on the moments of inertia and different objects, the moment of inertia with respect to the center of mass of a spherical shell is

I = 2/3 m R²

Let's calculate the moment of inertia of the sphere diameter (d = 2m8 and mass (m₁ = 1 kg)

r₁ = d / 2

r₁ = 2/2

r₁ = 1 m

I₁ = 2/3 m₁ r₁²

The other shell as a diameter (d = 1m)

r₂ = d / 2

r₂ = 1 / 2

r₂ = 0.5 m

I₂ = 2/3 m₂ r₂²

As they indicate that the two moments of inertia are equal

I₁ = I₂

m₁ r₁² = m₂ r₂²

m₂ = m₁ (r₁ / r₂)²

m₂ = 1 (1 / 0.5)²

m₂ = 4 kg

b) it is requested to find the relation of the moment of inertia of a disk and a sphere.

Moment of inertia disk I_disk = ½ m r²

Moment of inertia sphere I_ sphere = 2/5 m r²

The relationship between you is the division of moments of inertia

I_disk I _sphere = (1/2 m r²) / (2/5 m r²)

I_disk / I sphere = 5/4