Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Quick Summary
Gravitation explains the force that holds planets in orbit, keeps us on Earth, and determines tides. This chapter is important for NEET (space science questions) and forms the basis for Class 12 topics like electric fields.
Newton's Law of Gravitation
F = Gm₁m₂ / r²
- G = 6.674 × 10⁻¹¹ N m² kg⁻² (Universal gravitational constant)
- F acts along the line joining the two masses
- It is always attractive, acts at a distance, follows inverse square law
Acceleration Due to Gravity (g)
On Earth's surface: g = GM/R² ≈ 9.8 m s⁻²
Variation of g
| Factor | Formula | Effect |
|---|---|---|
| Altitude h | g' = g(1 − 2h/R) | g decreases |
| Depth d | g' = g(1 − d/R) | g decreases |
| Rotation of Earth | g_pole > g_equator | g is max at poles |
| At centre | g = 0 | zero |
Gravitational Field and Potential
- Gravitational field intensity: E = −GM/r² (vector, toward centre)
- Gravitational potential: V = −GM/r (scalar, negative)
- Gravitational potential energy: U = −GMm/r
Escape Velocity
Minimum speed to escape gravitational field:
v_e = √(2GM/R) = √(2gR)
For Earth: v_e ≈ 11.2 km/s
v_e = √2 × orbital velocity
Orbital Velocity and Satellites
For a satellite at height h from Earth's surface:
v₀ = √(GM/(R+h)) = √(gR²/(R+h))
At surface (h ≈ 0): v₀ = √(gR) ≈ 7.9 km/s
Time period: T = 2π(R+h)/v₀ = 2π√((R+h)³/GM)
Geostationary satellite:
- Time period = 24 hours
- Orbits at height ≈ 36,000 km above equator
- Remains stationary relative to Earth's surface
Kepler's Laws
1. Law of Orbits: All planets move in elliptical orbits with the Sun at one focus. 2. Law of Areas: The line joining a planet to the Sun sweeps equal areas in equal time (areal velocity is constant). 3. Law of Periods: T² ∝ a³ (square of orbital period proportional to cube of semi-major axis)
Important Questions
One Mark
- State Newton's law of gravitation.
- What is the value of G?
- Define escape velocity.
- What is a geostationary satellite?
- At what depth is g = 0?
Short Answer (3 marks)
- Derive the expression for escape velocity from Earth's surface.
- Explain how 'g' varies with altitude. Derive the expression.
- State Kepler's three laws of planetary motion.
Long Answer (5 marks)
- Derive the expression for orbital velocity of a satellite. Find the time period of revolution and derive Kepler's third law from Newton's law of gravitation.
- Explain gravitational field and potential. Derive the expression for gravitational potential energy of a system of two masses.
Common Mistakes
- Confusing escape velocity (√2gR) with orbital velocity (√gR).
- Using g as constant when finding potential energy far from Earth — use U = −GMm/r.
- Forgetting that gravitational potential is negative (the field does negative work to hold objects in orbit).
Board Exam Tips
- The derivation of escape velocity is a frequent 5-mark question — practise it fully.
- Kepler's laws conceptual questions appear regularly.
- NEET: satellite orbital speed and time period numericals appear almost every year.