A satellite is revolving round the earth in circular orbit

Options

A. if mass of earth is made four times, keeping other factors constant, orbital speed of satellite will become two times
B. corresponding to change in part(a), times period of satellite will become half
C. when value of `G` is made two times orbital speed increases and time period decreases
D. `G` has no effect on orbital speed and time period

Explanation

To solve the problem of a satellite revolving around the Earth in a circular orbit, we will follow these steps:

### Step 1: Understand the Forces Acting on the Satellite
A satellite in circular orbit experiences two main forces:
1. **Gravitational Force (F_gravity)**: This is the force exerted by the Earth on the satellite, given by the formula:
\[ F_{\text{gravity}} = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the distance from the center of the Earth to the satellite.

2. **Centripetal Force (F_centripetal)**: This is the force required to keep the satellite moving in a circular path, given by:
\[ F_{\text{centripetal}} = \frac{mv^2}{r} \]
where \( v \) is the orbital speed of the satellite.

### Step 2: Set the Forces Equal
For a satellite in a stable orbit, the gravitational force provides the necessary centripetal force. Therefore, we set the two forces equal:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \]

### Step 3: Simplify the Equation
We can cancel \( m \) (mass of the satellite) from both sides, as long as \( m \neq 0 \):
\[ \frac{GM}{r^2} = \frac{v^2}{r} \]
Multiplying both sides by \( r \) gives:
\[ \frac{GM}{r} = v^2 \]

### Step 4: Solve for Orbital Speed (v)
Taking the square root of both sides, we find the orbital speed \( v \):
\[ v = \sqrt{\frac{GM}{r}} \]

### Step 5: Determine the Time Period (T)
The time period \( T \) of the satellite's orbit can be calculated using the formula:
\[ T = \frac{2\pi r}{v} \]
Substituting the expression for \( v \) from Step 4:
\[ T = \frac{2\pi r}{\sqrt{\frac{GM}{r}}} \]
This simplifies to:
\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]

### Step 6: Analyze the Effects of Changes in Variables
1. If the mass of the satellite \( m \) is doubled, the orbital speed \( v \) does not change because it does not depend on \( m \).
2. If the gravitational constant \( G \) is increased, the orbital speed \( v \) increases, and the time period \( T \) decreases.
3. If the radius \( r \) increases, the orbital speed \( v \) decreases, and the time period \( T \) increases.

### Final Conclusion
From the analysis, we can conclude that:
- The orbital speed and time period are affected by changes in \( G \) and \( r \).
- The mass of the satellite does not affect the orbital speed or time period.