A sphere is rolling on a horizontal surface without slipping. The ratio of the rotational K.E. to the total kinetic energy of the sphere is

Options

A. `(2)/(5)`
B. `(2)/(7)`
C. `(5)/(7)`
D. `(3)/(7)`

Answer

To solve the problem of finding the ratio of the rotational kinetic energy to the total kinetic energy of a sphere rolling on a horizontal surface without slipping, we can follow these steps:

### Step 1: Understand the Kinetic Energies Involved
The total kinetic energy (TKE) of the sphere consists of two parts:
1. Translational Kinetic Energy (TKE_trans) due to its linear motion.
2. Rotational Kinetic Energy (TKE_rot) due to its rotation.

The formulas for these energies are:
- Translational Kinetic Energy:
\[ TKE_{\text{trans}} = \frac{1}{2} m v^2 \]
- Rotational Kinetic Energy:
\[ TKE_{\text{rot}} = \frac{1}{2} I \omega^2 \]

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

### Step 3: Relate Linear Velocity and Angular Velocity
Since the sphere rolls without slipping, the relationship between linear velocity (v) and angular velocity (ω) is:
\[ v = r \omega \]
Thus, we can express \( \omega \) in terms of \( v \):
\[ \omega = \frac{v}{r} \]

### Step 4: Substitute ω into the Rotational Kinetic Energy
Now we can substitute \( \omega \) into the equation for rotational kinetic energy:
\[ TKE_{\text{rot}} = \frac{1}{2} I \omega^2 = \frac{1}{2} \left(\frac{2}{5} m r^2\right) \left(\frac{v}{r}\right)^2 \]
This simplifies to:
\[ TKE_{\text{rot}} = \frac{1}{2} \cdot \frac{2}{5} m r^2 \cdot \frac{v^2}{r^2} = \frac{1}{5} m v^2 \]

### Step 5: Calculate Total Kinetic Energy
Now we can calculate the total kinetic energy:
\[ TKE_{\text{total}} = TKE_{\text{trans}} + TKE_{\text{rot}} = \frac{1}{2} m v^2 + \frac{1}{5} m v^2 \]
To combine these, we find a common denominator:
\[ TKE_{\text{total}} = \frac{5}{10} m v^2 + \frac{2}{10} m v^2 = \frac{7}{10} m v^2 \]

### Step 6: Find the Ratio of Rotational Kinetic Energy to Total Kinetic Energy
Now we can find the ratio of the rotational kinetic energy to the total kinetic energy:
\[ \text{Ratio} = \frac{TKE_{\text{rot}}}{TKE_{\text{total}}} = \frac{\frac{1}{5} m v^2}{\frac{7}{10} m v^2} \]
The \( m v^2 \) terms cancel out:
\[ \text{Ratio} = \frac{\frac{1}{5}}{\frac{7}{10}} = \frac{1}{5} \cdot \frac{10}{7} = \frac{2}{7} \]

### Final Answer
Thus, the ratio of the rotational kinetic energy to the total kinetic energy of the sphere is:
\[ \frac{2}{7} \]