A body weight 200 N on the surface of the earth . How much will it weight half way down to the centre of the earth

Options

A. 150 N
B. 200 N
C. 250 N
D. 100 N

Explanation

To solve the problem of how much a body weighing 200 N on the surface of the Earth will weigh halfway to the center of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Weight on the Surface:**
The weight of the body on the surface of the Earth is given as 200 N. Weight (W) is related to mass (m) and acceleration due to gravity (g) by the equation:
\[ W = m \cdot g \]
Here, we can express the weight as:
\[ W = 200 \, \text{N} \]

2. **Calculate the Mass of the Body:**
To find the mass of the body, we can rearrange the weight formula:
\[ m = \frac{W}{g} \]
On the surface of the Earth, \( g \) is approximately \( 9.81 \, \text{m/s}^2 \). Thus,
\[ m = \frac{200 \, \text{N}}{9.81 \, \text{m/s}^2} \approx 20.39 \, \text{kg} \]

3. **Determine the Acceleration Due to Gravity at Halfway Depth:**
The formula for the acceleration due to gravity at a depth \( d \) inside the Earth is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
Where \( R \) is the radius of the Earth. Since we are looking for the weight halfway to the center, \( d = \frac{R}{2} \):
\[ g' = g \left(1 - \frac{R/2}{R}\right) = g \left(1 - \frac{1}{2}\right) = \frac{g}{2} \]

4. **Calculate the New Weight at Halfway Depth:**
The new weight \( W' \) at this depth can be calculated using the new acceleration due to gravity:
\[ W' = m \cdot g' = m \cdot \frac{g}{2} \]
Substituting the mass we found earlier:
\[ W' = 20.39 \, \text{kg} \cdot \frac{9.81 \, \text{m/s}^2}{2} \approx 100 \, \text{N} \]

5. **Conclusion:**
Therefore, the weight of the body halfway down to the center of the Earth is approximately:
\[ W' \approx 100 \, \text{N} \]

### Final Answer:
The body will weigh **100 N** halfway down to the center of the Earth.

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