The force on a 1 kg mass on earth of radius R is 10 N. Then the force on a satellite revolving around the earth in the mean orbit 3R//2 will be (mass of satelite is 100 kg),

Options

A. `4.44xx10^(2)N`
B. `6.66xx10^(2)N`
C. `500N`
D. `3.33xx10^(2)N`

Answer

To solve the problem, we need to determine the gravitational force acting on a satellite of mass 100 kg that is orbiting the Earth at a distance of \( \frac{3R}{2} \) from the center of the Earth. We will use the formula for gravitational force:

\[ F = \frac{G M m}{r^2} \]

where:
- \( F \) is the gravitational force,
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the object (satellite),
- \( r \) is the distance from the center of the Earth to the object.

### Step-by-Step Solution:

1. **Identify the known values**:
- The force on a 1 kg mass at the surface of the Earth is given as \( F = 10 \, \text{N} \).
- The radius of the Earth is \( R \).
- The mass of the satellite is \( m = 100 \, \text{kg} \).
- The distance from the center of the Earth to the satellite is \( r = \frac{3R}{2} \).

2. **Use the gravitational force formula for the 1 kg mass**:
\[ F = \frac{G M (1)}{R^2} = 10 \, \text{N} \]
From this, we can express \( G M \):
\[ G M = 10 R^2 \]

3. **Calculate the gravitational force on the satellite**:
The gravitational force on the satellite at a distance of \( \frac{3R}{2} \) is given by:
\[ F_{\text{satellite}} = \frac{G M (100)}{\left(\frac{3R}{2}\right)^2} \]
Simplifying the denominator:
\[ \left(\frac{3R}{2}\right)^2 = \frac{9R^2}{4} \]
Thus, the force becomes:
\[ F_{\text{satellite}} = \frac{G M (100)}{\frac{9R^2}{4}} = \frac{400 G M}{9 R^2} \]

4. **Substitute \( G M \) from step 2**:
\[ F_{\text{satellite}} = \frac{400 (10 R^2)}{9 R^2} \]
The \( R^2 \) cancels out:
\[ F_{\text{satellite}} = \frac{4000}{9} \, \text{N} \]

5. **Calculate the numerical value**:
\[ F_{\text{satellite}} \approx 444.44 \, \text{N} \]

### Final Answer:
The force on the satellite is approximately \( 444.44 \, \text{N} \).

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