Kinetic energy distribution in a rolling object. Consider a hoop, a disk, and a sphere, each of mass M and radius R, that roll smoothly along a horizontal table. For each, what fraction of its kinetic energy is associated with the translation of its center of mass?

Explanation

To solve the problem of finding the fraction of kinetic energy associated with the translation of the center of mass for a hoop, a disk, and a sphere, we will follow these steps:

### Step 1: Define Kinetic Energy Components
The total kinetic energy (KE) of a rolling object is composed of two parts:
1. Translational Kinetic Energy (TKE): \( TKE = \frac{1}{2} M v^2 \)
2. Rotational Kinetic Energy (RKE): \( RKE = \frac{1}{2} I \omega^2 \)

Where:
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