Question

What will be the acceleration due to gravity at a depth \( d \), where \( g \) is the acceleration due to gravity on the surface of the Earth?

• A. \(\frac{g}{\left(1 + \frac{d}{R}\right)^2}\)
• B. \( g \left[ 1 - \frac{2d}{R} \right] \)
• C. \(\frac{g}{\left(1 - \frac{d}{R}\right)^2}\)
• D. \( g \left[ 1 - \frac{d}{R} \right] \)

Hint:

Acceleration Due to Gravity:

• At a height h above the Earth's surface: g' = g × (R / (R + h))²
• At a depth d below the Earth's surface: g' = g × (1 - d / R)
• Note: Gravity decreases linearly with depth inside the Earth.

Answer 1

The Correct Option is D

Step 1: Gravity Variation with Depth
The acceleration due to gravity at a depth d within the Earth is calculated using the formula:

g' = g × (1 - d / R)

Definitions:

• g': Acceleration due to gravity at depth d.
• g: Acceleration due to gravity at the Earth's surface.
• R: Radius of the Earth.

Step 2: Formula Justification
Gravitational force diminishes linearly with depth inside the Earth. This is because only the mass within a radius of R - d affects the gravitational force at depth d. This relationship is expressed as:

g' = g × (R - d) / R = g × (1 - d / R)

Therefore, the accurate equation is:

g' = g × (1 - d / R)

Selected Answer: Option (D)