A satellite is revolving near the earth.s surface. Its orbital velocity depends on its mass .m. as

Options

A. `m^0`
B. `m`
C. `m^2`
D. `m^3`

Answer

To solve the problem of determining how the orbital velocity of a satellite near the Earth's surface depends on its mass, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Satellite**:
- A satellite in orbit experiences two main forces: the gravitational force (Fg) pulling it towards the Earth and the centripetal force (Fc) required to keep it in circular motion.

2. **Write the Expression for Gravitational Force**:
- The gravitational force acting on the satellite can be expressed using Newton's law of gravitation:
\[ F_g = \frac{G M m}{r^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( r \) is the distance from the center of the Earth to the satellite (approximately equal to the radius of the Earth when the satellite is near the surface).

3. **Write the Expression for Centripetal Force**:
- The centripetal force required to keep the satellite in circular motion is given by:
\[ F_c = \frac{m v_0^2}{r} \]
where \( v_0 \) is the orbital velocity of the satellite.

4. **Set the Gravitational Force Equal to the Centripetal Force**:
- For a satellite in stable orbit, the gravitational force must equal the centripetal force:
\[ \frac{G M m}{r^2} = \frac{m v_0^2}{r} \]

5. **Cancel the Mass of the Satellite (m)**:
- Since the mass \( m \) of the satellite appears on both sides of the equation, we can cancel it out (assuming \( m \neq 0 \)):
\[ \frac{G M}{r^2} = \frac{v_0^2}{r} \]

6. **Rearrange the Equation to Solve for Orbital Velocity (v0)**:
- Multiply both sides by \( r \):
\[ \frac{G M}{r} = v_0^2 \]
- Taking the square root gives us the expression for the orbital velocity:
\[ v_0 = \sqrt{\frac{G M}{r}} \]

7. **Conclusion on the Dependence of Orbital Velocity on Mass**:
- From the derived expression \( v_0 = \sqrt{\frac{G M}{r}} \), we can see that the orbital velocity \( v_0 \) does not depend on the mass \( m \) of the satellite. It only depends on the mass \( M \) of the Earth and the radius \( r \).

### Final Answer:
The orbital velocity \( v_0 \) of a satellite near the Earth's surface is independent of its mass \( m \).