Question

The mass of the earth is 81 times the mass of the moon and the distance between their centres is \( R \). The distance from the centre of the earth where gravitational force will be zero is

• A. \( \frac{9R}{10} \)
• B. \( \frac{R}{2} \)
• C. \( \frac{R}{81} \)
• D. \( \frac{R}{4} \)

Hint:

When dealing with forces between two celestial bodies, the point where the forces balance is determined by the ratio of their masses and the distance between them.

Answer 1

The Correct Option is A

Step 1: Understanding the force equilibrium.
The point where the gravitational forces due to the earth and moon balance each other can be found using the formula: \[ \frac{GM_e}{x^2} = \frac{GM_m}{(R-x)^2} \] where \( G \) is the gravitational constant, \( M_e \) and \( M_m \) are the masses of the earth and moon, respectively, \( x \) is the distance from the earth, and \( R \) is the distance between the earth and the moon.

Step 2: Solving the equation.
Given that \( M_e = 81M_m \), the equation becomes: \[ \frac{81M_m}{x^2} = \frac{M_m}{(R-x)^2} \] Simplifying this yields: \[ x = \frac{9R}{10} \]
Step 3: Conclusion.
The correct answer is (A) \( \frac{9R}{10} \).