Find the weight of a body at a height of 100 km above the surface of earth. The radius of earth is 6,400 km and the body weighs 90 N on earth's surface.

Answer

To find the weight of a body at a height of 100 km above the surface of the Earth, we can follow these steps:

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

### Step 2: Calculate the mass of the body
Given that the weight of the body on the surface of the Earth is 90 N, we can find the mass \( m \) using the acceleration due to gravity at the Earth's surface, which is approximately \( g = 9.81 \, \text{m/s}^2 \):
\[ m = \frac{W}{g} = \frac{90 \, \text{N}}{9.81 \, \text{m/s}^2} \approx 9.17 \, \text{kg} \]

### Step 3: Determine the new acceleration due to gravity at height \( h \)
The formula for the acceleration due to gravity at a height \( h \) above the Earth's surface is:
\[ g_h = \frac{G M_e}{(R_e + h)^2} \]
where:
- \( G \) is the universal gravitational constant,
- \( M_e \) is the mass of the Earth,
- \( R_e \) is the radius of the Earth (6400 km),
- \( h \) is the height above the Earth's surface (100 km).

### Step 4: Substitute the values into the formula
Convert the radius of the Earth and the height into meters:
- \( R_e = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6.4 \times 10^6 \, \text{m} \)
- \( h = 100 \, \text{km} = 100 \times 10^3 \, \text{m} = 1 \times 10^5 \, \text{m} \)

Now substitute these values into the formula:
\[ g_h = \frac{G M_e}{(6.4 \times 10^6 + 1 \times 10^5)^2} \]
This simplifies to:
\[ g_h = \frac{G M_e}{(6.5 \times 10^6)^2} \]

### Step 5: Calculate the weight at height \( h \)
The new weight \( W_h \) at height \( h \) can be calculated using:
\[ W_h = m \cdot g_h \]
We know that:
\[ g_h = g \cdot \left( \frac{R_e^2}{(R_e + h)^2} \right) \]
Substituting the known values:
\[ W_h = m \cdot g \cdot \left( \frac{R_e^2}{(R_e + h)^2} \right) \]
Substituting \( m \approx 9.17 \, \text{kg} \), \( g \approx 9.81 \, \text{m/s}^2 \), \( R_e = 6400 \times 10^3 \, \text{m} \), and \( R_e + h = 6500 \times 10^3 \, \text{m} \):
\[ W_h = 9.17 \cdot 9.81 \cdot \left( \frac{(6400 \times 10^3)^2}{(6500 \times 10^3)^2} \right) \]

### Step 6: Calculate the final weight
Calculating the ratio:
\[ W_h \approx 90 \cdot \left( \frac{6400^2}{6500^2} \right) \approx 90 \cdot \left( \frac{40960000}{42250000} \right) \approx 90 \cdot 0.968 \approx 87.25 \, \text{N} \]

### Final Answer
The weight of the body at a height of 100 km above the surface of the Earth is approximately **87.25 N**.

---