Question

Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth \(d=\frac{R}{2}\) from the surface of earth, if its weight on the surface of earth is 200 N, will be 

• A. 100 N
• B. 300 N
• C. 50 N
• D. 150 N

Answer 1

The Correct Option is A

Step 1: Understanding the Problem 

The weight of a body decreases as it moves towards the center of the Earth because the effective gravitational force decreases. At a depth \( d \), the effective weight is proportional to the distance from the center of the Earth.

\( W_d = W_s \times \frac{R - d}{R} \)

Where:

• \( W_d \): Weight at depth \( d \)
• \( W_s = 200 \, \text{N} \): Weight at the surface
• \( R \): Radius of the Earth
• \( d = \frac{R}{2} \): Depth

Step 2: Substituting Values

From the formula: \[ W_d = W_s \times \frac{R - d}{R} \] Substituting \( W_s = 200 \, \text{N} \) and \( d = \frac{R}{2} \): \[ W_d = 200 \times \frac{R - \frac{R}{2}}{R} \]

Simplify the equation: \[ W_d = 200 \times \frac{\frac{R}{2}}{R} \] \[ W_d = 200 \times \frac{1}{2} = 100 \, \text{N} \]

Step 3: Final Answer

The weight of the body at depth \( d = \frac{R}{2} \) is \( \boxed{100 \, \text{N}} \).