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 1: Understand the forces acting on the satellite
A satellite in orbit experiences two primary forces:
- The gravitational force (Fg) acting towards the center of the Earth.
- The centripetal force (Fc) required to keep the satellite in circular motion.

### Step 2: Write down the expressions for gravitational and centripetal forces
The gravitational force acting on the satellite can be expressed as:
\[ F_g = \frac{G M m}{r^2} \]
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.

The centripetal force required to keep the satellite in circular motion is given by:
\[ F_c = \frac{m v^2}{r} \]
where:
- \( v \) is the orbital velocity of the satellite.

### Step 3: Set the gravitational force equal to the centripetal force
Since the satellite is in a stable orbit, the gravitational force must equal the centripetal force:
\[ \frac{G M m}{r^2} = \frac{m v^2}{r} \]

### Step 4: Simplify the equation
We can cancel the mass of the satellite \( m \) from both sides of the equation (assuming \( m \neq 0 \)):
\[ \frac{G M}{r^2} = \frac{v^2}{r} \]

### Step 5: Rearrange to find the expression for orbital velocity
Multiplying both sides by \( r \) gives:
\[ \frac{G M}{r} = v^2 \]
Taking the square root of both sides, we find the orbital velocity:
\[ v = \sqrt{\frac{G M}{r}} \]

### Step 6: Analyze the dependence on mass
From the derived expression for orbital velocity \( v \), we can see that it does not depend on the mass \( m \) of the satellite. The only variables that affect the orbital velocity are the gravitational constant \( G \), the mass of the Earth \( M \), and the radius \( r \).

### Conclusion
Thus, the orbital velocity of a satellite near the Earth's surface is independent of its mass.

### Final Answer
The orbital velocity depends on the mass of the satellite as \( v \propto m^0 \) (which means it does not depend on mass).

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