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A satellite is revolving near the earth....

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

A

`m^0`

B

m

C

`m^2`

D

`m^3`

Text Solution

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The correct Answer is:
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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