The weight of a dody at earth's surface is W. At a depth halfway to the centre of the earth , it will be (assuming uniform density in the earth )

Options

A. W
B. `W/2`
C. `W/4`
D. `W/8`

Answer

To solve the problem of determining the weight of a body at a depth halfway to the center of the Earth, we can follow these steps:

### Step 1: Understand the relationship between weight and gravitational acceleration
The weight of an object is given by the formula:
\[ W = mg \]
where \( W \) is the weight, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity.

### Step 2: Define the depth and radius
Let:
- \( R \) be the radius of the Earth.
- The depth halfway to the center of the Earth is \( d = \frac{R}{2} \).

### Step 3: Use the formula for gravitational acceleration at a depth
The acceleration due to gravity at a depth \( d \) below the Earth's surface is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where \( g \) is the acceleration due to gravity at the surface.

### Step 4: Substitute the depth into the formula
Substituting \( d = \frac{R}{2} \) into the equation:
\[ g' = g \left(1 - \frac{\frac{R}{2}}{R}\right) \]
This simplifies to:
\[ g' = g \left(1 - \frac{1}{2}\right) = g \left(\frac{1}{2}\right) = \frac{g}{2} \]

### Step 5: Calculate the new weight at the depth
Now, we can find the weight of the body at this depth:
\[ W' = mg' \]
Substituting \( g' \):
\[ W' = m \left(\frac{g}{2}\right) \]
Since the weight at the surface is \( W = mg \), we can express \( W' \) as:
\[ W' = \frac{1}{2} mg = \frac{W}{2} \]

### Conclusion
Thus, the weight of the body at a depth halfway to the center of the Earth will be:
\[ W' = \frac{W}{2} \]