If gravitational forces between a planet and a satellite is proportional to `R^(-5//2)`. If R is the orbit radius. Then the period of revolution of satellites is proportional to `R^(n)`. Find n.

Answer

To find the value of \( n \) in the relationship where the period of revolution of satellites is proportional to \( R^n \), given that the gravitational force is proportional to \( R^{-5/2} \), we can follow these steps:

### Step 1: Understand the relationship between gravitational force and centripetal force
The gravitational force \( F_g \) acting on the satellite is given by:
\[ F_g \propto \frac{M m}{R^{5/2}} \]
where \( M \) is the mass of the planet, \( m \) is the mass of the satellite, and \( R \) is the radius of the orbit.

### Step 2: Set up the centripetal force equation
The centripetal force required to keep the satellite in orbit is given by:
\[ F_c = \frac{m v^2}{R} \]
where \( v \) is the orbital velocity of the satellite.

### Step 3: Equate gravitational force to centripetal force
Setting the gravitational force equal to the centripetal force, we have:
\[ \frac{m v^2}{R} = G \frac{M m}{R^{5/2}} \]
Here, \( G \) is the gravitational constant.

### Step 4: Cancel the mass of the satellite
Since \( m \) appears on both sides, we can cancel it out:
\[ \frac{v^2}{R} = G \frac{M}{R^{5/2}} \]

### Step 5: Rearrange to find \( v^2 \)
Rearranging gives:
\[ v^2 = G M \frac{1}{R^{3/2}} \]

### Step 6: Find the expression for velocity \( v \)
Taking the square root:
\[ v = \sqrt{G M} R^{-3/4} \]

### Step 7: Use the formula for the period of revolution
The period \( T \) of revolution is given by:
\[ T = \frac{2 \pi R}{v} \]
Substituting the expression for \( v \):
\[ T = \frac{2 \pi R}{\sqrt{G M} R^{-3/4}} = \frac{2 \pi R^{1 + 3/4}}{\sqrt{G M}} = \frac{2 \pi R^{7/4}}{\sqrt{G M}} \]

### Step 8: Identify the proportionality
From the expression for \( T \), we can see that:
\[ T \propto R^{7/4} \]
Thus, we can conclude that:
\[ n = \frac{7}{4} \]

### Final Answer
The value of \( n \) is:
\[ \boxed{\frac{7}{4}} \]