Question

The value of acceleration due to gravity at a depth 'd' from the surface of earth and at an altitude 'h' from the surface of earth are in the ratio

• A. 1:1
• B. $\frac{R-2h}{R-d}$
• C. $\frac{R-d}{R-2h}$
• D. $\frac{R-d}{R-h}$

Hint:

Physics Tip : Note that the altitude formula $g_h = g(1-2h/R)$ is an approximation valid only when $h$ is much smaller than the Earth's radius $R$.

Answer 1

The Correct Option is C

Concept: Physics (Gravitation) - Variation of $g$ with Depth and Altitude.

Step 1: State the formula for $g$ at depth $d$. The acceleration due to gravity at a depth $d$ from the Earth's surface is given by: $$g_d = g \left[ 1 - \frac{d}{R} \right] = g \left[ \frac{R-d}{R} \right]$$

Step 2: State the formula for $g$ at altitude $h$. For small altitudes ($h \ll R$), the acceleration due to gravity at height $h$ is given by: $$g_h = g \left[ 1 - \frac{2h}{R} \right] = g \left[ \frac{R-2h}{R} \right]$$

Step 3: Calculate the ratio. Dividing the two expressions: $$\frac{g_d}{g_h} = \frac{g \left[ \frac{R-d}{R} \right]}{g \left[ \frac{R-2h}{R} \right]} = \frac{R-d}{R-2h}$$ $$ \therefore \text{The ratio of the acceleration due to gravity is } \frac{R-d}{R-2h}. \text{ } $$