Question

Acceleration due to gravity at a height of H from the surface of a plane is the same as that at a depth of H below the surface.If R is the radius of the planet, then H vs. R graph for different planets will be

• A.
• B.
• C.
• D.

Answer 1

The Correct Option is C

Relation Between Height and Depth for Equal Acceleration Due to Gravity 

The acceleration due to gravity at a height h above the Earth's surface is given by:

\[ g_h = g \left(1 - \frac{2h}{R}\right) \]

The acceleration due to gravity at a depth d below the Earth's surface is given by:

\[ g_d = g \left(1 - \frac{d}{R}\right) \]

To find the relation between height and depth when both give the same gravity, we equate \( g_h = g_d \):

\[ g \left(1 - \frac{2h}{R}\right) = g \left(1 - \frac{d}{R}\right) \]

Canceling \( g \) on both sides:

\[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \]

Simplifying:

\[ \frac{2h}{R} = \frac{d}{R} \Rightarrow 2h = d \]

Conclusion: When the acceleration due to gravity is the same at a height h and a depth d, then the relation is \( d = 2h \).

The correct answer is option (C):