Gravitation Exercise Solutions | KalviGavani

How to Use This Page

1. Attempt every exercise question on your own — then verify your method and answer with the solutions below. 2. Memorise the key formulas — gravitation has many interrelated formulas; understanding the derivations helps you recall them. 3. Pay special attention to numerical problems — they are the highest-mark questions in the exam. 4. Review the Common Mistakes section — these are the most frequent errors in this chapter. 5. Use the Exam Checklist to ensure you have covered every concept before the exam.

Key Concepts (Quick Recap)

Newton's Universal Law of Gravitation

F = Gm₁m₂/r²

Acceleration Due to Gravity

Kepler's Laws of Planetary Motion

Escape and Orbital Velocities

Satellites

Important Definitions

I. Choose the Best Answer (MCQs)

II. Fill in the Blanks

1. The universal gravitational constant G has the value ________ N·m²/kg². 6.674 × 10⁻¹¹ 2. The acceleration due to gravity at the centre of the Earth is ________. Zero 3. The escape velocity from a planet depends on its ________ and ________. Mass, radius 4. Kepler's first law states that planets move in ________ orbits. Elliptical 5. The relation between escape velocity and orbital velocity is vₑ = ________. √2 × vₒ 6. The value of g at the equator is ________ than at the poles. Less 7. The time period of a geostationary satellite is ________ hours. 24 8. If the mass of Earth were doubled and radius halved, the value of g would become ________ times. 8 9. The weight of a body in a freely falling lift is ________. Zero (weightlessness) 10. The orbital velocity of a satellite close to Earth's surface is approximately ________ km/s. 7.9 11. The gravitational force between two bodies is always ________ (attractive/repulsive). Attractive 12. The potential energy of a satellite is always ________ (positive/negative). Negative

III. True or False (with corrections)

1. The value of G changes with the distance between the bodies.

• False. G is a universal constant; its value (6.674 × 10⁻¹¹ N·m²/kg²) is the same everywhere in the universe and does not depend on the medium or distance.

2. The value of g is the same at all places on the surface of the Earth.

• False. g varies with latitude (maximum at poles, minimum at equator), altitude, and depth. It also varies with local geology.

3. A geostationary satellite can be placed at any latitude.

• False. A geostationary satellite must orbit in the equatorial plane, so it appears stationary relative to Earth. It cannot be placed at any latitude other than the equator.

4. Escape velocity depends on the mass of the escaping body.

• False. vₑ = √(2GM/R) depends on the mass M and radius R of the planet, not on the mass of the body escaping. A feather and a heavy rocket need the same escape velocity from the same planet.

5. If Earth stops rotating, the value of g at the equator would increase.

• True. The centrifugal effect due to rotation reduces g at the equator. If rotation stops, this reduction vanishes and g increases at the equator.

IV. Match the Following

V. Answer Briefly (2-mark Questions)

VI. Answer in Detail (5-mark Questions with Numerical Examples)

Common Mistakes to Avoid

1. Confusing G and g — G (6.674 × 10⁻¹¹ N·m²/kg²) is a universal constant; g (≈ 9.8 m/s²) is the acceleration due to gravity on Earth's surface and varies with location.

2. Forgetting to add height to Earth's radius — for a satellite at height h, the orbital radius is r = R + h, not just h. Always use r = R + h in orbital formulas.

3. Sign errors in potential energy — gravitational PE is negative (PE = −GMm/r). The total energy TE = −GMm/2r is also negative for bound orbits. A positive total energy means the body is not bound.

4. Using g = 9.8 m/s² at all altitudes — g decreases with altitude. Use g_h = GM/(R+h)² or the approximation g(1 − 2h/R) for h << R.

5. Not converting units — distances in km must be converted to m before using in formulas with G. Heights given in km → multiply by 10³.

6. Mixing up escape and orbital velocities — vₑ = √(2GM/R) and vₒ = √(GM/R). Remember vₑ = √2 × vₒ. They are not the same.

7. Applying depth formula for altitude (or vice versa) — for altitude: g_h ≈ g(1 − 2h/R); for depth: g_d = g(1 − d/R). These are different formulas.

8. Forgetting that gravitational force is always attractive — it can never be repulsive. The negative sign in PE indicates attraction, not repulsion.

Exam Checklist

• [ ] Can state Newton's law of gravitation and derive g = GM/R²
• [ ] Know how g varies with altitude (g_h ≈ g(1 − 2h/R)) and depth (g_d = g(1 − d/R))
• [ ] Can state and explain all three of Kepler's laws
• [ ] Can derive escape velocity (vₑ = √(2GM/R)) and orbital velocity (vₒ = √(GM/r))
• [ ] Know the relation vₑ = √2 × vₒ and can derive it
• [ ] Can calculate time period of a satellite (T = 2π√(r³/GM))
• [ ] Know the energy of a satellite: KE = GMm/2r, PE = −GMm/r, TE = −GMm/2r
• [ ] Can explain weightlessness and when it occurs
• [ ] Remember unit conversions (km to m) and G = 6.674 × 10⁻¹¹ N·m²/kg²
• [ ] Can explain why g is greater at poles than at equator (shape + rotation)