Gravitation – IIT/JEE Solved Problems
Phase 1 • Page 4 • Satellites, Time Period & Energy Change
Problem 16: Time Period Ratio of Satellites
Q: Two satellites revolve around Earth in circular orbits of radii R and 8R. Find the ratio of their time periods.
Solution:
Time period of satellite:
T ∝ r3/2
T2 / T1 = (8R / R)3/2 = 83/2 = 16√2
IIT Shortcut: Never derive formula again—remember T ∝ r3/2.
Problem 17: Satellite Speed Ratio
Q: Find the ratio of orbital speeds of two satellites at radii R and 4R.
Solution:
Orbital speed:
v ∝ 1/√r
v1 / v2 = √(4R / R) = 2
Concept: Higher orbit ⇒ lower speed.
Problem 18: Energy Change on Moving to Higher Orbit
Q: A satellite is moved from orbit of radius r to 2r. Find change in total energy.
Solution:
Total energy:
E = −GMm / (2r)
Initial energy:
E1 = −GMm / (2r)
Final energy:
E2 = −GMm / (4r)
Change in energy:
ΔE = E2 − E1 = +GMm / (4r)
Important: Energy increases (becomes less negative) when satellite goes to higher orbit.
Problem 19: Satellite Falling Towards Earth
Q: If a satellite loses some energy due to friction, what happens to its orbit?
Answer:
Loss of energy causes satellite to move to a lower orbit. Orbital radius decreases and speed increases. Eventually, it may burn due to atmospheric friction.
IIT Insight: Lower orbit = lower energy but higher speed.
Problem 20: Satellite at Geostationary Orbit
Q: Why must a geostationary satellite revolve in equatorial plane?
Answer:
For a satellite to remain fixed over a point on Earth, its orbital plane must coincide with Earth’s equatorial plane.
Key Point: Geostationary satellites are special circular orbits.
Phase-1 High-Value Relations
- T ∝ r3/2
- v ∝ 1/√r
- E ∝ −1/r
- Higher orbit ⇒ lower speed, higher energy
Phase 1 • Page 4 Completed Successfully ✅
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- Gravitation – Part 8
- Gravitation – Part 9
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- Gravitation – Part 17
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- Gravitation – Part 19
- Gravitation – Part 20
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- Gravitation – Part 35
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- Gravitation – Part 40
- Gravitation – Part 41
- Gravitation – Part 42
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