Question

If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gravity at a depth \(\alpha h \ (h \ll Re)\) from the earth surface. The value of α will be ________.
(Use R_e=6400 km)

Answer 1

Correct Answer: 2

To determine the value of α where the gravitational acceleration at height h above Earth's surface equals the gravitational acceleration at a depth αh below the surface, we proceed with the following analysis: 

1. The acceleration due to gravity at height h is given by:
   \( g_h = g_0 \left(1 - \frac{2h}{R_e}\right) \)
2. At a depth d, the gravity becomes:
   \( g_d = g_0 \left(1 - \frac{d}{R_e}\right) \)
3. According to the problem, these accelerations are equal, i.e.,
   \(\left(1 - \frac{2h}{R_e}\right) = \left(1 - \frac{\alpha h}{R_e}\right) \)
4. Solving for α:
   • Start by equating the gravitational terms:
     \( 1 - \frac{2h}{R_e} = 1 - \frac{\alpha h}{R_e} \)
   • Cancel out the common terms and rearrange:
     \( \frac{2h}{R_e} = \frac{\alpha h}{R_e} \)
   • Since h/R_e is common, we simplify:
     \( 2 = \alpha \)

The calculated value of α is 2, matching the specified range of 2 to 2, thereby confirming the solution is accurate.