The geostationalry orbit oif the earth is at a distance of about 36000 km from the earth's surface. Find the weight of a 120 kg equipment placed in a geostationary satellite. The radius of the earth is 6400 km.

Options

A. `1/2h`
B. `1h`
C. `2h`
D. `4h`

Explanation

To find the weight of a 120 kg equipment placed in a geostationary satellite, we will follow these steps:

### Step 1: Understand the problem
We need to calculate the weight of an object in a geostationary orbit, which is located 36,000 km above the Earth's surface. The radius of the Earth is given as 6400 km.

### Step 2: Calculate the total distance from the center of the Earth
The total distance (r) from the center of the Earth to the satellite is the sum of the Earth's radius and the height of the satellite above the Earth's surface.

\[ r = R + h \]
Where:
- \( R = 6400 \, \text{km} \)
- \( h = 36000 \, \text{km} \)

Calculating \( r \):
\[ r = 6400 \, \text{km} + 36000 \, \text{km} = 42400 \, \text{km} = 42400 \times 10^3 \, \text{m} = 42400000 \, \text{m} \]

### Step 3: Calculate the gravitational intensity (g) at the height of the satellite
The formula for gravitational intensity at a distance \( r \) from the center of the Earth is given by:

\[ g' = \frac{GM}{r^2} \]
Where \( G \) is the gravitational constant and \( M \) is the mass of the Earth. However, we can use the ratio of gravitational intensities at the surface and at height \( h \):

\[ \frac{g'}{g} = \frac{R^2}{r^2} \]

Where \( g \) is the acceleration due to gravity at the surface of the Earth (approximately \( 9.8 \, \text{m/s}^2 \)).

### Step 4: Substitute the values into the equation
Using the values we have:
- \( R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)
- \( r = 42400 \, \text{km} = 42400 \times 10^3 \, \text{m} \)

Calculating \( g' \):
\[ g' = g \cdot \frac{R^2}{r^2} \]
\[ g' = 9.8 \cdot \frac{(6400 \times 10^3)^2}{(42400 \times 10^3)^2} \]

### Step 5: Calculate the values
Calculating \( g' \):
\[ g' = 9.8 \cdot \frac{(6400)^2}{(42400)^2} \]
\[ g' = 9.8 \cdot \frac{40960000}{1797760000} \approx 0.22 \, \text{m/s}^2 \]

### Step 6: Calculate the weight of the equipment
The weight \( W \) of the equipment is given by:
\[ W = m \cdot g' \]
Where \( m = 120 \, \text{kg} \).

Calculating \( W \):
\[ W = 120 \cdot 0.22 = 26.4 \, \text{N} \]

### Final Answer
The weight of the 120 kg equipment in a geostationary satellite is approximately **26.4 Newtons**.

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