Two point masses of mass `4m` and `m` respectively separated by `d` distance are revolving under mutual force of attraction. Ration of their kinetic energies will be:

Options

A. `1:4`
B. `4:1`
C. `1:1`
D. `1:16`

Answer

To solve the problem of finding the ratio of the kinetic energies of two point masses \(4m\) and \(m\) that are revolving under mutual gravitational attraction, we can follow these steps:

### Step 1: Determine the Center of Mass
The center of mass (CM) for two point masses can be calculated using the formula:

\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]

Here, let \(m_1 = 4m\) (mass at position 0) and \(m_2 = m\) (mass at position \(d\)).

\[ x_{CM} = \frac{4m \cdot 0 + m \cdot d}{4m + m} = \frac{md}{5m} = \frac{d}{5} \]

### Step 2: Calculate Distances from the Center of Mass
Now, we can find the distances of each mass from the center of mass:

- Distance of \(4m\) from CM: \(x_{4m} = \frac{d}{5}\)
- Distance of \(m\) from CM: \(x_m = d - \frac{d}{5} = \frac{4d}{5}\)

### Step 3: Write the Expression for Centripetal Force
For circular motion, the centripetal force acting on each mass is given by:

\[ F_c = m \cdot r \cdot \omega^2 \]

Where \(r\) is the distance from the center of mass and \(\omega\) is the angular velocity.

For mass \(4m\):
\[ F_{c, 4m} = 4m \cdot \frac{d}{5} \cdot \omega^2 \]

For mass \(m\):
\[ F_{c, m} = m \cdot \frac{4d}{5} \cdot \omega^2 \]

### Step 4: Set Forces Equal
Since both masses are in circular motion due to their mutual gravitational attraction, we can set the centripetal forces equal to each other:

\[ 4m \cdot \frac{d}{5} \cdot \omega^2 = m \cdot \frac{4d}{5} \cdot \omega^2 \]

### Step 5: Calculate Kinetic Energies
The kinetic energy (\(KE\)) for each mass in rotational motion is given by:

\[ KE = \frac{1}{2} I \omega^2 \]

Where \(I\) is the moment of inertia. The moment of inertia for point masses is given by \(I = m \cdot r^2\).

For mass \(4m\):
\[ I_{4m} = 4m \cdot \left(\frac{d}{5}\right)^2 = 4m \cdot \frac{d^2}{25} = \frac{4md^2}{25} \]
\[ KE_{4m} = \frac{1}{2} \cdot \frac{4md^2}{25} \cdot \omega^2 = \frac{2md^2 \omega^2}{25} \]

For mass \(m\):
\[ I_m = m \cdot \left(\frac{4d}{5}\right)^2 = m \cdot \frac{16d^2}{25} = \frac{16md^2}{25} \]
\[ KE_m = \frac{1}{2} \cdot \frac{16md^2}{25} \cdot \omega^2 = \frac{8md^2 \omega^2}{25} \]

### Step 6: Calculate the Ratio of Kinetic Energies
Now, we can find the ratio of the kinetic energies:

\[ \text{Ratio} = \frac{KE_{4m}}{KE_m} = \frac{\frac{2md^2 \omega^2}{25}}{\frac{8md^2 \omega^2}{25}} = \frac{2}{8} = \frac{1}{4} \]

Thus, the ratio of the kinetic energies of the two masses is:

\[ \text{Ratio of Kinetic Energies} = 1 : 4 \]

### Final Answer
The ratio of their kinetic energies is \(1 : 4\).

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