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

To solve this problem, we need to understand the relationship between the angular momentum of a satellite and its orbit around the Earth.

The orbital angular momentum \( L \) of a satellite revolving around the Earth is given by the formula:

L = m \cdot v \cdot r

where:

• m is the mass of the satellite.
• v is the orbital velocity of the satellite.
• r is the distance from the center of the Earth to the satellite (orbital radius).

Initially, if the orbital angular momentum is \( L \) and the distance from the Earth's center to the satellite is \( r \), then:

L = m \cdot v \cdot r

Now, the distance from the Earth's center is increased by eight times its initial value, which means the new radius is \( 8r \).

The new angular momentum \( L_{\text{new}} \) will be:

L_{\text{new}} = m \cdot v_{\text{new}} \cdot (8r).

For circular orbits, the centripetal force required for keeping the satellite in orbit is provided by the gravitational force. Therefore, using the equation for circular orbits:

\frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r}

Solving for \( v \), we get v = \sqrt{\frac{G \cdot M}{r}}.

If the distance becomes \( 8r \), the new velocity \( v_{\text{new}} \) becomes:

v_{\text{new}} = \sqrt{\frac{G \cdot M}{8r}} = \frac{1}{\sqrt{8}}v = \frac{1}{2\sqrt{2}} v.

So, the new angular momentum is:

L_{\text{new}} = m \cdot \left(\frac{1}{2\sqrt{2}}v\right) \cdot 8r = \frac{8}{2\sqrt{2}} \cdot m \cdot v \cdot r = \frac{8}{2\sqrt{2}} \cdot L = \sqrt{2} \cdot L = \frac{8}{4} L

This simplifies to:

L_{\text{new}} = 2\sqrt{2} \cdot L \approx 3 \cdot L

Therefore, the new angular momentum is approximately 3 times the initial angular momentum, which matches the option 3L.