Question

The distance between two stars of masses 3M_S and 6M_S is 9R. Here R is the mean distance between the centres of the Earth and the Sun, and M_S is the mass of the Sun. The two stars orbit around their common centre of mass in circular orbits with period nT, where T is the period of Earth’s revolution around the Sun. The value of n is ___.

Answer 1

Correct Answer: 9

Step 1: Understanding the system
We are given a two-body system where the centre of mass lies at a distance of 6R from the lighter mass. The system is rotating under gravitational interaction.

Step 2: Apply centripetal force balance for heavier mass
Assume the heavier mass is 3M_s and the lighter is M_s.
Distance of 3M_s from centre of mass = R
Distance of M_s from centre of mass = 6R
Using centripetal force formula for heavier mass:
\( 3M_s \omega^2 \cdot 6R = \frac{G \cdot 3M_s \cdot M_s}{(R + 6R)^2} = \frac{3GM_s^2}{49R^2} \)
The correct distance between them is 7R, hence the denominator is (7R)² = 49R².

Step 3: Cancel mass and simplify
\( \omega^2 = \frac{G M_s}{(49R^2) \cdot (18R)} = \frac{GM_s}{81R^3} \)

Step 4: Use ω to find time period
We know the relation: \( T = \frac{2\pi}{\omega} \Rightarrow T' = 2\pi \sqrt{\frac{1}{\omega^2}} \)
So,
\( T' = \sqrt{\frac{81R^3}{GM_s}} \)

Step 5: Compare with standard time period
If the standard time period is \( T = \sqrt{\frac{R^3}{GM_s}} \)
Then,
\( T' = 9T \) ⇒ n = 9

Final Answer:
The value of n = 9