A disc is rolling on a surface without slipping. What is the ratio of its translational to rotational kinetic energies ?

Options

A. `5 : 2`
B. `2: 1`
C. `3 : 2`
D. `2 : 3`

Answer

To find the ratio of translational to rotational kinetic energies for a disc rolling without slipping, we can follow these steps:

### Step 1: Define the Kinetic Energies
The translational kinetic energy (TKE) of the disc is given by the formula:
\[ \text{TKE} = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the disc and \( v \) is the velocity of the center of mass.

The rotational kinetic energy (RKE) of the disc is given by the formula:
\[ \text{RKE} = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

### Step 2: Determine the Moment of Inertia
For a solid disc, the moment of inertia about its center is:
\[ I = \frac{1}{2} m r^2 \]
where \( r \) is the radius of the disc.

### Step 3: Relate Linear and Angular Velocity
For a disc rolling without slipping, the relationship between linear velocity \( v \) and angular velocity \( \omega \) is:
\[ v = r \omega \]
From this, we can express \( \omega \) in terms of \( v \):
\[ \omega = \frac{v}{r} \]

### Step 4: Substitute \( \omega \) in RKE
Now, substituting \( \omega \) in the expression for rotational kinetic energy:
\[ \text{RKE} = \frac{1}{2} I \omega^2 = \frac{1}{2} \left(\frac{1}{2} m r^2\right) \left(\frac{v}{r}\right)^2 \]
This simplifies to:
\[ \text{RKE} = \frac{1}{2} \left(\frac{1}{2} m r^2\right) \left(\frac{v^2}{r^2}\right) = \frac{1}{4} m v^2 \]

### Step 5: Calculate the Ratio of TKE to RKE
Now we can find the ratio of translational kinetic energy to rotational kinetic energy:
\[ \text{Ratio} = \frac{\text{TKE}}{\text{RKE}} = \frac{\frac{1}{2} m v^2}{\frac{1}{4} m v^2} \]
The \( m v^2 \) terms cancel out:
\[ \text{Ratio} = \frac{\frac{1}{2}}{\frac{1}{4}} = \frac{1/2}{1/4} = 2 \]

### Conclusion
Thus, the ratio of translational to rotational kinetic energies for a disc rolling without slipping is:
\[ \text{Ratio} = 2:1 \]