A circular disc rolls down an inclined plane . The ratio of rotational kinetic energy to total kinetic energy is

Options

A. `1/2`
B. `1/3`
C. `2/3`
D. `3/4`

Answer

To find the ratio of rotational kinetic energy to total kinetic energy for a circular disc rolling down an inclined plane, we can follow these steps:

### Step 1: Define Rotational Kinetic Energy
The rotational kinetic energy (KE_rot) of a circular disc is given by the formula:
\[ KE_{\text{rot}} = \frac{1}{2} I \omega^2 \]
where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

### Step 2: Find the Moment of Inertia for a Disc
For a solid circular disc, the moment of inertia \(I\) is:
\[ I = \frac{1}{2} m r^2 \]
where \(m\) is the mass of the disc and \(r\) is its radius.

### Step 3: Substitute the Moment of Inertia into the Rotational Kinetic Energy Formula
Substituting \(I\) into the rotational kinetic energy formula:
\[ KE_{\text{rot}} = \frac{1}{2} \left( \frac{1}{2} m r^2 \right) \omega^2 = \frac{1}{4} m r^2 \omega^2 \]

### Step 4: Define Translational Kinetic Energy
The translational kinetic energy (KE_trans) is given by:
\[ KE_{\text{trans}} = \frac{1}{2} m v^2 \]
where \(v\) is the linear velocity of the center of mass of the disc.

### Step 5: Relate Linear Velocity to Angular Velocity
For a disc rolling without slipping, the relationship between linear velocity \(v\) and angular velocity \(\omega\) is:
\[ v = r \omega \]
Substituting this into the translational kinetic energy formula:
\[ KE_{\text{trans}} = \frac{1}{2} m (r \omega)^2 = \frac{1}{2} m r^2 \omega^2 \]

### Step 6: Calculate Total Kinetic Energy
The total kinetic energy (KE_total) is the sum of rotational and translational kinetic energies:
\[ KE_{\text{total}} = KE_{\text{rot}} + KE_{\text{trans}} = \frac{1}{4} m r^2 \omega^2 + \frac{1}{2} m r^2 \omega^2 \]
To combine these, convert \(\frac{1}{2}\) to a fraction with a common denominator:
\[ KE_{\text{total}} = \frac{1}{4} m r^2 \omega^2 + \frac{2}{4} m r^2 \omega^2 = \frac{3}{4} m r^2 \omega^2 \]

### Step 7: Find the Ratio of Rotational Kinetic Energy to Total Kinetic Energy
Now we can find the ratio:
\[ \text{Ratio} = \frac{KE_{\text{rot}}}{KE_{\text{total}}} = \frac{\frac{1}{4} m r^2 \omega^2}{\frac{3}{4} m r^2 \omega^2} \]
The \(m\), \(r^2\), and \(\omega^2\) terms cancel out:
\[ \text{Ratio} = \frac{1/4}{3/4} = \frac{1}{3} \]

### Conclusion
The ratio of rotational kinetic energy to total kinetic energy for a circular disc rolling down an inclined plane is:
\[ \frac{1}{3} \]