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If the time of revolution of a satellite is T, then the potential energy will be proportional to:
- a)T1/3
- b)T3
- c)T-2/3
- d)T-4/3
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If the time of revolution of a satellite is T, then the potential ener...
Kepler's third law relates the period (T) of an orbiting body to its semi-major axis (a). The potential energy is inversely proportional to T2/3, as per Kepler's third law.
Most Upvoted Answer
If the time of revolution of a satellite is T, then the potential ener...
Explanation:
Relation between potential energy and time of revolution:
- The potential energy of a satellite in orbit is given by the formula:
\( PE = - \frac{G M m}{2r} \)
- Where PE is potential energy, G is the gravitational constant, M is the mass of the planet, m is the mass of the satellite, and r is the distance between the satellite and the center of the planet.
Relation between time period and distance:
- The time period of revolution of a satellite in circular orbit is given by:
\( T = 2 \pi \sqrt{\frac{r^3}{G M}} \)
- Where T is the time period, r is the distance between the satellite and the center of the planet, G is the gravitational constant, and M is the mass of the planet.
Substitute the value of r in the potential energy formula:
- From the equation for time period, we can express r in terms of T as:
\( r = (\frac{G M T^2}{4 \pi^2})^{1/3} \)
- Substitute this value of r in the potential energy formula:
\( PE = - \frac{G M m}{2} \times \frac{4 \pi^2}{G M T^2} \)
\( PE = - 2 \pi^2 \frac{m}{T^2} \)
Conclusion:
- The potential energy is inversely proportional to the square of the time period of revolution. Therefore, the potential energy is proportional to \( T^{-2} \), which corresponds to option 'C'.
Relation between potential energy and time of revolution:
- The potential energy of a satellite in orbit is given by the formula:
\( PE = - \frac{G M m}{2r} \)
- Where PE is potential energy, G is the gravitational constant, M is the mass of the planet, m is the mass of the satellite, and r is the distance between the satellite and the center of the planet.
Relation between time period and distance:
- The time period of revolution of a satellite in circular orbit is given by:
\( T = 2 \pi \sqrt{\frac{r^3}{G M}} \)
- Where T is the time period, r is the distance between the satellite and the center of the planet, G is the gravitational constant, and M is the mass of the planet.
Substitute the value of r in the potential energy formula:
- From the equation for time period, we can express r in terms of T as:
\( r = (\frac{G M T^2}{4 \pi^2})^{1/3} \)
- Substitute this value of r in the potential energy formula:
\( PE = - \frac{G M m}{2} \times \frac{4 \pi^2}{G M T^2} \)
\( PE = - 2 \pi^2 \frac{m}{T^2} \)
Conclusion:
- The potential energy is inversely proportional to the square of the time period of revolution. Therefore, the potential energy is proportional to \( T^{-2} \), which corresponds to option 'C'.
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