Find the percentage decrease in the weight of a body when taken 16 km below the surface of the earth. Take radius of the earth is 6400 km.

Answer

To find the percentage decrease in the weight of a body when taken 16 km below the surface of the Earth, we can follow these steps:

### Step 1: Understand the relationship between weight and gravity
The weight of a body at the surface of the Earth is given by:
\[ W = mg \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity at the surface.

### Step 2: Determine the formula for gravity below the Earth's surface
When a body is taken below the Earth's surface, the acceleration due to gravity \( g' \) can be calculated using the formula:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where:
- \( d \) is the depth below the surface (16 km in this case),
- \( R \) is the radius of the Earth (6400 km).

### Step 3: Substitute the values into the formula
Substituting the values into the formula:
\[ g' = g \left(1 - \frac{16}{6400}\right) \]
Calculating \( \frac{16}{6400} \):
\[ \frac{16}{6400} = \frac{1}{400} \]
Thus, we can rewrite the equation as:
\[ g' = g \left(1 - \frac{1}{400}\right) \]
\[ g' = g \left(\frac{399}{400}\right) \]

### Step 4: Calculate the new weight
The weight of the body at 16 km below the surface is:
\[ W' = mg' = m \left(g \frac{399}{400}\right) = \frac{399}{400} mg \]
Thus, the weight at this depth is:
\[ W' = \frac{399}{400} W \]

### Step 5: Calculate the decrease in weight
The decrease in weight \( \Delta W \) can be calculated as:
\[ \Delta W = W - W' = W - \frac{399}{400} W = W \left(1 - \frac{399}{400}\right) \]
\[ \Delta W = W \left(\frac{1}{400}\right) \]

### Step 6: Calculate the percentage decrease in weight
The percentage decrease in weight is given by:
\[ \text{Percentage decrease} = \frac{\Delta W}{W} \times 100 \]
Substituting the value of \( \Delta W \):
\[ \text{Percentage decrease} = \frac{\frac{1}{400} W}{W} \times 100 = \frac{1}{400} \times 100 \]
\[ \text{Percentage decrease} = \frac{100}{400} = 0.25\% \]

### Final Answer
The percentage decrease in the weight of the body when taken 16 km below the surface of the Earth is **0.25%**.

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