Two satellites of masses M and 4M are orbiting the earth in a circular orbit of radius r. Their frequencies of revolution are in the ratio of

Options

A. `1 : 4`
B. `4 : 1`
C. `1 : 2`
D. `1 : 1`

Explanation

To find the ratio of the frequencies of revolution of two satellites with masses \( M \) and \( 4M \) orbiting the Earth at a radius \( r \), we can follow these steps:

### Step 1: Understand the relationship between frequency and orbital radius
The frequency of revolution \( f \) of a satellite in a circular orbit is related to the gravitational force acting on it and the centripetal force required to keep it in orbit. The gravitational force provides the necessary centripetal force for circular motion.

### Step 2: Write down the gravitational force
The gravitational force \( F_g \) acting on a satellite of mass \( m \) at a distance \( r \) from the center of the Earth is given by:
\[ F_g = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

### Step 3: Write down the centripetal force
The centripetal force \( F_c \) required to keep the satellite in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the orbital speed of the satellite.

### Step 4: Set the gravitational force equal to the centripetal force
For a satellite in a stable orbit, the gravitational force equals the centripetal force:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \]

### Step 5: Simplify the equation
We can cancel \( m \) from both sides (since \( m \neq 0 \)):
\[ \frac{GM}{r^2} = \frac{v^2}{r} \]
Multiplying both sides by \( r \):
\[ \frac{GM}{r} = v^2 \]

### Step 6: Find the orbital speed
Taking the square root gives us the orbital speed \( v \):
\[ v = \sqrt{\frac{GM}{r}} \]

### Step 7: Relate frequency to orbital speed
The frequency \( f \) of revolution is related to the orbital speed and the circumference of the orbit:
\[ f = \frac{v}{2\pi r} \]
Substituting for \( v \):
\[ f = \frac{\sqrt{\frac{GM}{r}}}{2\pi r} = \frac{1}{2\pi} \sqrt{\frac{GM}{r^3}} \]

### Step 8: Determine the ratio of frequencies
Since the frequency \( f \) is independent of the mass of the satellite, both satellites (mass \( M \) and \( 4M \)) will have the same frequency:
\[ f_1 = \frac{1}{2\pi} \sqrt{\frac{GM}{r^3}} \quad \text{(for mass \( M \))} \]
\[ f_2 = \frac{1}{2\pi} \sqrt{\frac{GM}{r^3}} \quad \text{(for mass \( 4M \))} \]

Thus, the ratio of their frequencies is:
\[ \frac{f_1}{f_2} = 1 \]

### Final Answer
The ratio of the frequencies of revolution of the two satellites is \( 1:1 \).

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