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: Understanding Gravity Variation with Depth
The acceleration due to gravity at a depth d inside the Earth is given by the formula:

g' = g × (1 - d / R)

Where:

• g' is the acceleration due to gravity at depth d.
• g is the acceleration due to gravity on the surface of the Earth.
• R is the radius of the Earth.

Step 2: Explanation of the Formula
Gravity decreases linearly as we go deeper inside the Earth because only the mass enclosed within the radius R - d contributes to the gravitational force at depth d. This leads to the relation:

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

Thus, the correct expression is:

g' = g × (1 - d / R)

Correct Option: Option (D)

Answer 2

To determine the acceleration due to gravity at a depth \( d \) beneath the Earth's surface, we start with the understanding that gravity decreases as we move inside the Earth. The Earth's gravitational force at a depth \( d \) with radius \( R \) can be expressed using the formula: \( g_d = g \left[ 1 - \frac{d}{R} \right] \). This follows from the assumption that inside the Earth, gravitational force relates directly to the distance from the center:

• Step 1: Consider that on the Earth's surface, the gravitational force is \( g \).
• Step 2: At a depth \( d \), the effective radius contributing to gravity decreases: the effective radius becomes \( R - d \).
• Step 3: The gravitational acceleration at this depth is thus proportionally reduced as only the mass between the center and the depth contributes: \( g_d = g\frac{R-d}{R} \).
• The relation simplifies to \( g_d = g \left[ 1 - \frac{d}{R} \right] \), showcasing a linear decrease in gravitational force with depth.

This confirms that the correct formula for gravity at depth is given by \( g \left[ 1 - \frac{d}{R} \right] \).