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

Explanation

To solve the problem of how much a body weighing 250 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 250 N. This weight is due to the gravitational force acting on the body, which can be expressed as:
\[ W = mg \]
where \( W \) is the weight, \( m \) is the mass of the body, and \( g \) is the acceleration due to gravity on the surface of the Earth.

2. **Calculate the Mass of the Body**:
To find the mass \( m \), we can rearrange the formula:
\[ m = \frac{W}{g} \]
Assuming \( g \approx 9.8 \, \text{m/s}^2 \) (the average acceleration due to gravity at the surface of the Earth), we can calculate:
\[ m = \frac{250 \, \text{N}}{9.8 \, \text{m/s}^2} \approx 25.51 \, \text{kg} \]

3. **Determine the Depth**:
The problem states that we need to find the weight halfway to the center of the Earth. The depth \( d \) at this point is half the radius of the Earth, which we denote as:
\[ d = \frac{R}{2} \]
where \( R \) is the radius of the Earth.

4. **Calculate the Acceleration Due to Gravity at Depth**:
The formula for the acceleration due to gravity at a depth \( d \) is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
Substituting \( d = \frac{R}{2} \):
\[ g' = g \left(1 - \frac{R/2}{R}\right) = g \left(1 - \frac{1}{2}\right) = \frac{g}{2} \]
Thus, at halfway to the center of the Earth, the acceleration due to gravity is:
\[ g' = \frac{g}{2} \approx \frac{9.8 \, \text{m/s}^2}{2} \approx 4.9 \, \text{m/s}^2 \]

5. **Calculate the Weight at Halfway Depth**:
Now, we can find the new weight \( W' \) of the body at this depth using the formula:
\[ W' = mg' \]
Substituting \( m \) and \( g' \):
\[ W' = 25.51 \, \text{kg} \times 4.9 \, \text{m/s}^2 \approx 125 \, \text{N} \]

### Final Answer:
The weight of the body halfway down to the center of the Earth is approximately **125 N**.