Question

Two stars of masses \( M \) and \( 2M \) that are at a distance \( d \) apart, are revolving one around another. The angular velocity of the system of two stars is:

• A. \( \sqrt{\frac{4GM}{d^3}} \)
• B. \( \sqrt{\frac{2GM}{d^3}} \)
• C. \( \sqrt{\frac{9GM}{d^3}} \)
• D. \( \sqrt{\frac{3GM}{d^3}} \) \bigskip

Hint:

For two stars revolving around each other due to gravitational attraction, the angular velocity of the system can be determined using Newton's law of gravitation and the concept of centripetal force.

Answer 1

The Correct Option is D

For two stars revolving around each other due to gravitational attraction, the angular velocity of the system can be determined using Newton's law of gravitation and the concept of centripetal force. Step 1: Gravitational Force The gravitational force between two masses \( M \) and \( 2M \) is given by: \[ F = \frac{G \cdot M \cdot 2M}{d^2} = \frac{2GM^2}{d^2} \] where \( G \) is the gravitational constant and \( d \) is the distance between the two masses. Step 2: Centripetal Force For a circular orbit, the centripetal force for each mass is given by: \[ F = M \cdot \omega^2 \cdot \frac{d}{2} \] where \( \omega \) is the angular velocity, and the factor of \( \frac{d}{2} \) is used because the center of mass of the system is at the midpoint between the two masses. Step 3: Equating Forces Now, equate the gravitational force and centripetal force: \[ \frac{2GM^2}{d^2} = M \cdot \omega^2 \cdot \frac{d}{2} \] Solving for \( \omega^2 \): \[ \omega^2 = \frac{4GM}{d^3} \] Thus, the angular velocity \( \omega \) is: \[ \omega = \sqrt{\frac{4GM}{d^3}} \]