A body weighs 200 N on the surface of the earth. How much will it weigh 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 down 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. 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 at the surface of the Earth.

2. **Determine the Mass of the Body**:
To find the mass \( m \), we can rearrange the weight equation:
\[ m = \frac{W}{g} \]
However, we do not need to calculate \( m \) explicitly since we will be using the ratio of weights.

3. **Calculate the Change in Acceleration due to Gravity**:
As we move inside the Earth, the acceleration due to gravity changes. 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 \( g' \) is the acceleration due to gravity at depth \( d \), \( g \) is the acceleration due to gravity at the surface, and \( R \) is the radius of the Earth.

4. **Determine the Depth**:
Since we are going halfway to the center of the Earth, the depth \( d \) is:
\[ d = \frac{R}{2} \]

5. **Substitute the Depth into the Gravity Formula**:
Substitute \( d = \frac{R}{2} \) into the gravity formula:
\[ g' = g \left(1 - \frac{\frac{R}{2}}{R}\right) = g \left(1 - \frac{1}{2}\right) = g \left(\frac{1}{2}\right) = \frac{g}{2} \]

6. **Calculate the Weight at Halfway Depth**:
The weight of the body at this new depth can be calculated using the new acceleration due to gravity:
\[ W' = mg' = m \left(\frac{g}{2}\right) \]
Since we know \( W = mg \) and \( W' = \frac{1}{2} W \):
\[ W' = \frac{1}{2} \times 200 \text{ N} = 100 \text{ N} \]

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

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