A body weighs 250N on the surface of the earth. How much will it weighs half way down to the centre of the earth?

Options

A. `125N`
B. `150N`
C. `175N`
D. `250N`

Explanation

To find the weight of a body halfway to the center of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Weight of the Body**:
The weight \( W \) of a body is given by the formula:
\[ W = m \cdot g \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity at the surface of the Earth.

2. **Given Information**:
The weight of the body on the surface of the Earth is given as \( W = 250 \, \text{N} \).

3. **Calculate the Mass of the Body**:
To find the mass \( m \), we can rearrange the weight formula:
\[ m = \frac{W}{g} \]
At the surface of the Earth, \( g \approx 9.81 \, \text{m/s}^2 \). Therefore:
\[ m = \frac{250 \, \text{N}}{9.81 \, \text{m/s}^2} \approx 25.5 \, \text{kg} \]

4. **Determine the Change in Acceleration Due to Gravity**:
Inside the Earth, the acceleration due to gravity \( g' \) at a distance \( d \) from the center can be calculated using the formula:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where \( R \) is the radius of the Earth and \( d \) is the distance from the center.

5. **Calculate \( g' \) Halfway to the Center**:
Halfway to the center means \( d = \frac{R}{2} \). Thus:
\[ g' = g \left(1 - \frac{\frac{R}{2}}{R}\right) = g \left(1 - \frac{1}{2}\right) = \frac{g}{2} \]
Therefore, \( g' = \frac{9.81 \, \text{m/s}^2}{2} \approx 4.905 \, \text{m/s}^2 \).

6. **Calculate the Weight at Halfway**:
Now, we can find the weight of the body halfway to the center:
\[ W' = m \cdot g' = 25.5 \, \text{kg} \cdot 4.905 \, \text{m/s}^2 \approx 125 \, \text{N} \]

7. **Final Answer**:
The weight of the body halfway down to the center of the Earth is approximately \( 125 \, \text{N} \).