A disk and a ring of the same mass are rolling to have the same kinetic energy. What is ratio of their velocities of centre of mass

Options

A. `(4:3)^(1//2)`
B. `(3:4)^(1//2)`
C. `(2)^(1//2):(3)^(1//2)`
D. `(3)^(1//2):(2)^(1//2)`

Explanation

To solve the problem of finding the ratio of the velocities of the center of mass of a disk and a ring of the same mass rolling with the same kinetic energy, we can follow these steps:

### Step 1: Define the Kinetic Energy for Both Objects
The total kinetic energy (KE) of a rolling object can be expressed as the sum of translational and rotational kinetic energy:
\[ KE = \frac{1}{2} m v_{cm}^2 + \frac{1}{2} I \omega^2 \]
where:
- \(m\) is the mass,
- \(v_{cm}\) is the velocity of the center of mass,
- \(I\) is the moment of inertia,
- \(\omega\) is the angular velocity.

### Step 2: Moment of Inertia for Disk and Ring
For a disk of mass \(m\) and radius \(r_d\):
\[ I_d = \frac{1}{2} m r_d^2 \]
For a ring of mass \(m\) and radius \(r_r\):
\[ I_r = m r_r^2 \]

### Step 3: Relate Angular Velocity to Linear Velocity
For rolling without slipping, the relationship between linear velocity and angular velocity is given by:
\[ v_{cm} = r \omega \]
Thus:
- For the disk: \(\omega_d = \frac{v_d}{r_d}\)
- For the ring: \(\omega_r = \frac{v_r}{r_r}\)

### Step 4: Write the Kinetic Energy Equations
For the disk:
\[ KE_d = \frac{1}{2} m v_d^2 + \frac{1}{2} \left(\frac{1}{2} m r_d^2\right) \left(\frac{v_d}{r_d}\right)^2 \]
Simplifying this:
\[ KE_d = \frac{1}{2} m v_d^2 + \frac{1}{4} m v_d^2 = \frac{3}{4} m v_d^2 \]

For the ring:
\[ KE_r = \frac{1}{2} m v_r^2 + \frac{1}{2} \left(m r_r^2\right) \left(\frac{v_r}{r_r}\right)^2 \]
Simplifying this:
\[ KE_r = \frac{1}{2} m v_r^2 + \frac{1}{2} m v_r^2 = m v_r^2 \]

### Step 5: Set the Kinetic Energies Equal
Since the kinetic energies are the same:
\[ \frac{3}{4} m v_d^2 = m v_r^2 \]
Cancelling \(m\) from both sides:
\[ \frac{3}{4} v_d^2 = v_r^2 \]

### Step 6: Solve for the Ratio of Velocities
Taking the square root of both sides gives:
\[ \frac{v_d}{v_r} = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \]

### Final Answer
Thus, the ratio of the velocities of the center of mass of the disk to the ring is:
\[ \frac{v_d}{v_r} = \frac{2}{\sqrt{3}} \]