Question

The Sun rotates around its centre once in 27 days. What will be the period of revolution if the Sun were to expand to twice its present radius without any external influence? Assume the Sun to be a sphere of uniform density.

• A. \( 108 \text{ days} \)
• B. \( 115 \text{ days} \)
• C. \( 100 \text{ days} \)
• D. \( 54 \text{ days} \)

Hint:

Remember the conservation of angular momentum \(I_1 \omega_1 = I_2 \omega_2\). The moment of inertia of a uniform sphere is \(I = \frac{2}{5} MR^2\). If the radius changes and mass remains constant, the moment of inertia changes with the square of the radius.

Answer 1

The Correct Option is A

To calculate the new period of revolution if the Sun's radius doubles, we will apply the principle of conservation of angular momentum. The angular momentum \( L \) of a rotating body is defined as:

\( L = I \omega \)

where \( I \) denotes the moment of inertia and \( \omega \) represents the angular velocity. For a sphere of uniform density, the moment of inertia \( I \) is given by:

\( I = \frac{2}{5}MR^2 \)

where \( M \) is the mass and \( R \) is the radius. Due to the absence of external torque, angular momentum remains constant throughout the expansion process:

\( I_1 \omega_1 = I_2 \omega_2 \)

The initial state is described by:

\( I_1 = \frac{2}{5}MR_1^2 \)

Following expansion, the radius becomes \( 2R_1 \), leading to:

\( I_2 = \frac{2}{5}M(2R_1)^2 = \frac{8}{5}MR_1^2 \)

Given that the initial rotational period \( T_1 \) is 27 days, the initial angular velocity is:

\( \omega_1 = \frac{2\pi}{T_1} \)

The new angular velocity \( \omega_2 \) is expressed as:

\( \omega_2 = \frac{2\pi}{T_2} \)

Applying conservation of angular momentum yields:

\( \frac{2}{5}MR_1^2 \cdot \frac{2\pi}{27} = \frac{8}{5}MR_1^2 \cdot \frac{2\pi}{T_2} \)

After canceling common factors \( \frac{2\pi}{5}MR_1^2 \), the equation simplifies to:

\( \frac{1}{27} = \frac{4}{T_2} \)

Solving for \( T_2 \):

\( T_2 = 4 \times 27 = 108 \text{ days} \)

Therefore, a doubling of the Sun's radius results in a revolution period of \( 108 \text{ days} \).