Determine the decrease in the weight of a body when it is taken 32 km below the earth surface. Take radius of the earth as 6400 km.

Options

A. 1%
B. 0.5%
C. 2%
D. NONE OF THE ABOVE

Explanation

To determine the decrease in the weight of a body when it is taken 32 km below the Earth's surface, we can follow these steps:

### Step 1: Understand the relationship between weight and gravitational acceleration
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.

### Step 2: Calculate the gravitational acceleration at the surface of the Earth
At the surface of the Earth, the gravitational acceleration is denoted by \( g \).

### Step 3: Use the formula for gravitational acceleration at a depth
The gravitational acceleration \( g_d \) at a depth \( d \) below the Earth's surface can be calculated using the formula:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]
where:
- \( d \) is the depth (32 km in this case),
- \( R \) is the radius of the Earth (6400 km).

### Step 4: Substitute the values into the formula
Substituting \( d = 32 \) km and \( R = 6400 \) km into the formula:
\[ g_d = g \left(1 - \frac{32}{6400}\right) \]
Calculating \( \frac{32}{6400} \):
\[ \frac{32}{6400} = \frac{1}{200} \]
Thus, we have:
\[ g_d = g \left(1 - \frac{1}{200}\right) = g \left(\frac{199}{200}\right) \]

### Step 5: Calculate the weight at the depth
The weight of the body at the depth \( d \) is:
\[ W_d = m \cdot g_d = m \cdot g \left(\frac{199}{200}\right) \]

### Step 6: Calculate the decrease in weight
The weight at the surface is:
\[ W_s = m \cdot g \]
The decrease in weight \( \Delta W \) when the body is taken to depth \( d \) is:
\[ \Delta W = W_s - W_d = m \cdot g - m \cdot g \left(\frac{199}{200}\right) \]
Factoring out \( m \cdot g \):
\[ \Delta W = m \cdot g \left(1 - \frac{199}{200}\right) = m \cdot g \left(\frac{1}{200}\right) \]

### Step 7: Calculate the percentage decrease in weight
The percentage decrease in weight can be calculated as:
\[ \text{Percentage Decrease} = \frac{\Delta W}{W_s} \times 100 = \frac{m \cdot g \left(\frac{1}{200}\right)}{m \cdot g} \times 100 = \frac{1}{200} \times 100 = 0.5\% \]

### Final Answer
The decrease in the weight of the body when taken 32 km below the Earth's surface is **0.5%**.