A solid sphere of radius `r` is rolling on a horizontal surface. The ratio between the rotational kinetic energy and total energy.

Options

A. `(5)/(7)`
B. `(2)/(7)`
C. `(1)/(2)`
D. `(1)/(7)`

Explanation

To find the ratio between the rotational kinetic energy and the total energy of a solid sphere rolling on a horizontal surface, we can follow these steps:

### Step 1: Understand the Kinetic Energies
A solid sphere rolling without slipping has two types of kinetic energy:
1. **Translational Kinetic Energy (TKE)**: This is due to the motion of the center of mass and is given by:
\[ \text{TKE} = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the sphere and \( v \) is the translational velocity of the center of mass.

2. **Rotational Kinetic Energy (RKE)**: This is due to the rotation about its center of mass and is given by:
\[ \text{RKE} = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

### Step 2: Calculate the Moment of Inertia
For a solid sphere, the moment of inertia \( I \) about its center of mass is:
\[ I = \frac{2}{5} m r^2 \]
where \( r \) is the radius of the sphere.

### Step 3: Relate Angular Velocity and Translational Velocity
For rolling without slipping, the relationship between translational velocity \( v \) and angular velocity \( \omega \) is:
\[ v = \omega r \quad \Rightarrow \quad \omega = \frac{v}{r} \]

### Step 4: Substitute Angular Velocity into RKE
Substituting \( \omega \) into the equation for RKE:
\[ \text{RKE} = \frac{1}{2} I \omega^2 = \frac{1}{2} \left(\frac{2}{5} m r^2\right) \left(\frac{v}{r}\right)^2 = \frac{1}{2} \cdot \frac{2}{5} m r^2 \cdot \frac{v^2}{r^2} = \frac{1}{5} m v^2 \]

### Step 5: Calculate Total Energy
The total energy \( E \) of the rolling sphere is the sum of translational and rotational kinetic energies:
\[ E = \text{TKE} + \text{RKE} = \frac{1}{2} m v^2 + \frac{1}{5} m v^2 \]
To combine these, we need a common denominator:
\[ E = \frac{5}{10} m v^2 + \frac{2}{10} m v^2 = \frac{7}{10} m v^2 \]

### Step 6: Find the Ratio of RKE to Total Energy
Now, we can find the ratio of RKE to total energy:
\[ \text{Ratio} = \frac{\text{RKE}}{E} = \frac{\frac{1}{5} m v^2}{\frac{7}{10} m v^2} \]
Cancelling \( m v^2 \) from the numerator and denominator:
\[ \text{Ratio} = \frac{1/5}{7/10} = \frac{1}{5} \cdot \frac{10}{7} = \frac{2}{7} \]

### Final Answer
The ratio between the rotational kinetic energy and the total energy of a solid sphere rolling on a horizontal surface is:
\[ \frac{2}{7} \]