Chapter 7: Gravitation
Original notes — ready CBSE (NCERT, English Medium) • Class 11 • Physics • Book: Physics — Part I
Gravitation — EG's own summary, in simple words
Book: Physics — Part I (CBSE, Class 11) | Chapter 7 | Medium: English
Gravitation is the weakest force in everyday experience and the one that shapes the universe. Newton's law of universal gravitation explains falling apples and orbiting planets with one rule, and Kepler's laws describe how planets actually move.
Diagram (labels on the image): The Moon orbits the Earth because gravity pulls it towards the centre while its forward motion keeps it falling around, never into, the Earth.
What you should learn
- State Newton's law of universal gravitation and explain the inverse-square dependence.
- State Kepler's three laws and connect the second to angular momentum conservation.
- Derive g = GM/R² and explain how g varies with height and depth.
- Use gravitational potential energy U = −GMm/r and explain its negative sign.
- Compare escape speed with orbital speed and describe geostationary and polar satellites.
1. Newton's law of universal gravitation
Every mass attracts every other mass with a force F = G m₁m₂ / r², where r is the distance between their centres and G is the universal gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²). The force is attractive, acts along the line joining the masses, and is equal and opposite on the two bodies. It is an inverse-square law: doubling the distance cuts the force to one quarter. The law is universal — it works for an apple, the Moon and distant stars alike.
2. Acceleration due to gravity (g)
Near the Earth's surface, gravity gives every freely falling body an acceleration g = GM/R², about 9.8 m/s², where M is Earth's mass and R its radius. Notice that g does not depend on the falling body's mass — a heavy stone and a light stone fall together (ignoring air). g is slightly smaller at the equator than at the poles (Earth bulges at the equator) and it decreases with height above the surface and with depth below it.
3. Kepler's laws of planetary motion
Kepler found three laws from observations before Newton explained them. First, planets move in ellipses with the Sun at one focus. Second, a line joining a planet to the Sun sweeps out equal areas in equal times — so a planet moves faster when nearer the Sun. Third, the square of a planet's orbital period is proportional to the cube of its mean distance from the Sun (T² ∝ r³). Newton's gravitation explains all three: the second law is really conservation of angular momentum.
4. Gravitational potential energy and escape speed
Gravitational potential energy is zero at infinity and negative elsewhere: U = −GMm/r. The negative sign means energy must be supplied to separate the masses. Escape speed is the minimum speed needed to escape a planet's gravity and never return: v-esc = √(2GM/R) = √(2gR), about 11.2 km/s for Earth. A rocket does not need to reach it all at once in practice, but the value sets the energy scale for leaving Earth.
5. Satellites
A satellite is a projectile falling around the Earth: it keeps missing the ground because of its forward speed. Orbital speed near Earth's surface is v = √(gR) ≈ 7.9 km/s, and the period is about 84 minutes for a low orbit. Geostationary satellites orbit above the equator with a 24-hour period, so they appear fixed in the sky — ideal for communication and TV. Weightlessness in orbit is not the absence of gravity; it is free fall, with satellite and astronaut falling together.
Full coverage — every section of this chapter
§7.1 Introduction: Gravitation is the force that holds planets in orbit and pulls a thrown ball back down. The chapter's achievement is one law for both: the same gravity that drops an apple holds the Moon.
§7.2 Kepler's laws: First, planets move in ellipses with the Sun at one focus. Second, the line joining a planet to the Sun sweeps equal areas in equal times, so a planet moves faster near the Sun — this is conservation of angular momentum in disguise. Third, the square of the period is proportional to the cube of the mean distance, T² ∝ r³.
§7.3 Universal law of gravitation: Every mass attracts every other with F = G m₁m₂/r² along the line joining them. The force is mutual and equal on both bodies, and for several bodies the net force is the vector sum of the individual attractions.
§7.4 The gravitational constant: G = 6.67 × 10⁻¹¹ N m² kg⁻², measured by Cavendish with a torsion balance. Gravity is so weak that measuring G needs a delicate experiment: the attraction between laboratory masses is tiny, which is why G was measured more than a century after Newton stated the law.
§7.5 Acceleration due to gravity of the earth: Near the surface, g = GM/R² ≈ 9.8 m/s², independent of the falling body's mass. The same formula gives g on any planet from its mass and radius.
§7.6 Acceleration due to gravity below and above the surface of earth: Above the surface, g(h) = g(R/(R + h))², approximately g(1 − 2h/R) for heights small compared with R. Below the surface, g(d) = g(1 − d/R) for depth d, falling linearly to zero at the centre for a uniform Earth.
§7.7 Gravitational potential energy: With zero fixed at infinity, U = −GMm/r. The sign is negative because gravity is attractive: energy must be supplied to separate the masses to infinity. Near the surface this reduces to the familiar mgh, measured from an arbitrary local reference level.
§7.8 Escape speed: The minimum speed to escape to infinity and never return is v_e = √(2GM/R) = √(2gR), about 11.2 km/s from Earth. It depends on the planet, not on the escaping body's mass or on the direction of projection (air resistance aside).
§7.9 Earth satellites: A satellite in circular orbit at radius r has orbital speed v = √(GM/r) and period T = 2π√(r³/GM). Geostationary satellites orbit in the equatorial plane with a 24-hour period, appearing fixed above one point; polar satellites pass over both poles and scan the whole Earth as it turns beneath them. An orbiting astronaut feels weightless because both astronaut and spacecraft fall freely together.
§7.10 Energy of an orbiting satellite: For a circular orbit, kinetic energy is +GMm/2r and potential energy −GMm/r, so total energy is E = −GMm/2r. The negative total energy is what it means to be bound: supplying +GMm/2r of energy frees the satellite to infinity.
All ten sections above are numbered sections of the rationalised chapter; this page carries no legacy topic.
Chart / Table — Gravity numbers worth memorising
| Quantity | Symbol / formula | Earth value |
|---|---|---|
| Gravitational constant | G | 6.67 × 10⁻¹¹ N·m²/kg² |
| Surface gravity | g = GM/R² | ≈ 9.8 m/s² |
| Low-orbit speed | √(gR) | ≈ 7.9 km/s |
| Escape speed | √(2gR) | ≈ 11.2 km/s |
| Geostationary period | T | 24 hours |
Caption: A tree with an apple beneath it. The same gravity that pulls an apple down holds the Moon in its orbit — one law for the falling apple and the circling Moon.
Practice set — try it yourself
- Why does doubling the distance between two masses reduce the gravitational force to one quarter? Hint: Look at how r appears in the formula.
- A stone and a feather are dropped together on the Moon (no air). Which lands first? Hint: Compare their accelerations, not their masses.
- Use Kepler's third law: a planet twice as far from the Sun as Earth has what orbital period? Hint: T² ∝ r³; Earth has T = 1 year, r = 1 unit.
- Calculate the escape speed from Earth using g = 10 m/s² and R = 6400 km. Hint: v = √(2gR); watch the units of R.
- Why does a geostationary satellite have to orbit above the equator? Hint: Think about the plane of its orbit and Earth's rotation.
Key Terms / Important Terms
- Gravitation — the universal attraction between masses.
- Gravitational constant (G) — 6.67 × 10⁻¹¹ N·m²/kg², the same everywhere in the universe.
- Acceleration due to gravity (g) — GM/R² near a planet's surface; ≈ 9.8 m/s² on Earth.
- Kepler's laws — three laws describing planetary orbits: ellipse, equal areas, T² ∝ r³.
- Gravitational potential energy — −GMm/r, zero at infinity.
- Escape speed — minimum speed to escape a planet's gravity: √(2gR).
- Orbital speed — speed needed for a stable circular orbit: √(GM/r).
- Geostationary satellite — a satellite with a 24-hour period above the equator, appearing fixed.
- Weightlessness — the feeling of free fall, when gravity acts but no support force is felt.
- Polar satellite — a satellite in a low orbit passing over both poles, which scans the whole Earth as the planet turns beneath it.
- Torsion balance — Cavendish's delicate instrument for measuring the tiny gravitational attraction between laboratory masses, giving G.
- Binding energy — the energy that must be supplied to free a bound satellite to infinity: +GMm/2r for a circular orbit, the magnitude of its negative total energy.
Remember
- F = G m₁m₂/r² — inverse square: double distance, quarter force.
- g does not depend on the falling body's mass.
- Kepler 2 (equal areas) = angular momentum conservation; Kepler 3: T² ∝ r³.
- Escape speed from Earth ≈ 11.2 km/s; low-orbit speed ≈ 7.9 km/s.
- Weightlessness in orbit is free fall, not zero gravity.
- Escape speed depends on the PLANET (its M and R), never on the mass of the escaping body — a stone and a rocket need the same 11.2 km/s from Earth.
Exam Tip / Common Mistake
Common mistake: saying there is no gravity in orbit. Gravity in low orbit is still about 90% of surface gravity — astronauts float because they are in continuous free fall around the Earth, not because gravity has vanished. If gravity were absent, the satellite would fly off in a straight line.
Questions with Answers — full answers
- State Newton's law of universal gravitation and write its formula. Answer: Every body attracts every other body with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them: F = G m₁m₂/r².
- Why do all bodies fall with the same acceleration near Earth's surface, regardless of mass? Answer: Gravitational force on a body is proportional to its mass (F = GMm/R²), but acceleration is force divided by mass, so mass cancels: a = GM/R², the same for all bodies.
- A planet is 4 times as far from the Sun as Earth. What is its orbital period in Earth years? Answer: T² ∝ r³, so T² = 4³ = 64, giving T = 8 years.
- Distinguish orbital speed and escape speed for Earth. Answer: Orbital speed (≈ 7.9 km/s) keeps a satellite in a low circular orbit. Escape speed (≈ 11.2 km/s, √2 times larger) lets a body leave Earth's gravity entirely.
- An astronaut in an orbiting station floats. Is gravity acting on her? Explain. Answer: Yes, gravity is still strong in orbit. She floats because she and the station are both in free fall around the Earth, so there is no support force pushing on her feet.
Extension Questions with Answers — go further
- What is g at a height equal to Earth's radius above the surface? Answer: g(h) = g(R/(R + h))² = g(1/2)² = g/4, about 2.45 m/s².
- A planet's mean distance from the Sun doubles. By what factor does its orbital period change? Answer: By Kepler's third law, T² ∝ r³, so T grows by 2√2 ≈ 2.83 times.
- Why is a satellite's total energy negative, and what does that mean? Answer: Because E = K + U = GMm/2r − GMm/r = −GMm/2r. Negative total energy means the satellite is bound: it cannot reach infinity without added energy.
- Why does a geostationary satellite have to orbit over the equator? Answer: Its orbit must be centred on Earth's centre with a 24-hour period to stay fixed above one point, and only an equatorial orbit keeps it above the same spot as Earth turns.
- An astronaut in orbit floats and feels weightless. Has gravity stopped acting on them? Answer: No. Both astronaut and spacecraft are in free fall together under gravity, so there is no normal reaction to feel as weight.
Board-pattern Questions with Answers
- State Newton's law of universal gravitation and give the value of G. Answer: F = G m₁m₂/r², acting along the line joining the masses; G = 6.67 × 10⁻¹¹ N m² kg⁻².
- Derive g = GM/R² for a body near Earth's surface. Answer: Gravity on mass m is GMm/R², and by Newton's second law this equals mg, so cancelling m gives g = GM/R².
- State Kepler's third law. Answer: The square of a planet's orbital period is proportional to the cube of its mean distance from the Sun: T² ∝ r³.
- Derive the escape speed from Earth in terms of g and R. Answer: Setting kinetic energy equal to the energy needed to reach infinity, ½mv² = GMm/R, gives v = √(2GM/R) = √(2gR) ≈ 11.2 km/s.
- What is g at a depth of R/2 below Earth's surface? Answer: g(d) = g(1 − d/R) = g(1 − 1/2) = g/2, about 4.9 m/s².
Recap — the chapter in five lines
- One inverse-square law governs apples, the Moon and the planets.
- Kepler's laws describe the motion; Newton's gravitation explains them.
- g = GM/R² falls with height as (R/(R+h))² and with depth as (1 − d/R).
- Potential energy is negative because gravity must be fought to escape.
- Bound orbits have negative total energy; escape speed is √(2gR).