What is the ratio of the weights of a body when it is kept at a height 500m above the surface of the earth and 500m below the surface of the earth, if the radius of the earth is 6400km.

Explanation

To find the ratio of the weights of a body when it is kept at a height of 500 meters above the surface of the Earth and 500 meters below the surface of the Earth, we can follow these steps:

### Step 1: Understand the Weight of the Body
The weight of a body (W) 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 Acceleration due to Gravity at Height
At a height \( h \) above the Earth's surface, the acceleration due to gravity \( g_1 \) is given by:
\[ g_1 = g \cdot \frac{R}{R + h} \]
where:
- \( g \) is the acceleration due to gravity at the Earth's surface (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( R \) is the radius of the Earth (6400 km or \( 6400 \times 10^3 \, \text{m} \)),
- \( h \) is the height above the surface (500 m).

### Step 3: Substitute Values for Height
Substituting the values into the formula:
\[ g_1 = g \cdot \frac{6400 \times 10^3}{6400 \times 10^3 + 500} \]

### Step 4: Calculate the Acceleration due to Gravity at Depth
At a depth \( d \) below the Earth's surface, the acceleration due to gravity \( g_2 \) is given by:
\[ g_2 = g \cdot \left(1 - \frac{d}{R}\right) \]
where \( d \) is the depth (500 m).

### Step 5: Substitute Values for Depth
Substituting the values into the formula:
\[ g_2 = g \cdot \left(1 - \frac{500}{6400 \times 10^3}\right) \]

### Step 6: Calculate the Ratio of Weights
The ratio of the weights \( W_1 \) (at height) and \( W_2 \) (at depth) can be expressed as:
\[ \frac{W_1}{W_2} = \frac{g_1}{g_2} \]

### Step 7: Substitute the Expressions for \( g_1 \) and \( g_2 \)
Thus, we have:
\[ \frac{W_1}{W_2} = \frac{g \cdot \frac{R}{R + h}}{g \cdot \left(1 - \frac{d}{R}\right)} \]

### Step 8: Simplify the Expression
The mass \( m \) cancels out, leading to:
\[ \frac{W_1}{W_2} = \frac{R}{R + h} \cdot \frac{1}{1 - \frac{d}{R}} \]

### Step 9: Substitute the Known Values
Now substituting \( R = 6400 \times 10^3 \, \text{m} \), \( h = 500 \, \text{m} \), and \( d = 500 \, \text{m} \):
\[ \frac{W_1}{W_2} = \frac{6400 \times 10^3}{6400 \times 10^3 + 500} \cdot \frac{1}{1 - \frac{500}{6400 \times 10^3}} \]

### Step 10: Calculate the Final Ratio
After performing the calculations, you will find that:
\[ \frac{W_1}{W_2} \approx 0.99 \]

### Final Answer
The ratio of the weights of the body at a height of 500 m above the surface and 500 m below the surface is approximately \( 0.99 \).

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