Gravitation covers Newton’s universal law of attraction (F = G m1 m2 / r squared), how the acceleration due to gravity g changes with height and depth, and the orbital and escape velocities that govern satellites and rockets. It also includes Kepler’s three laws of planetary motion and gravitational potential and potential energy. It is a steady scoring NEET chapter, rich in direct formula-based questions on g variation, escape velocity and Kepler’s third law.
Table of Contents
- Key Concepts - Kepler’s laws, universal gravitation, g and its variation, potential energy, escape and orbital velocity, satellites
- Weightage in Board & Entrance Exams
- Important Definitions
- Solved Examples
- Important Questions for Board Exams
- Quick Revision Points
Key Concepts
1. Kepler’s Laws of Planetary Motion
Before Newton explained why planets move, Johannes Kepler described how they move using three laws based on Tycho Brahe’s observations.
- First law (Law of Orbits): Every planet moves around the Sun in an elliptical orbit with the Sun at one focus.
- Second law (Law of Areas): The line joining a planet to the Sun sweeps out equal areas in equal intervals of time. This means a planet moves faster when nearer the Sun and slower when farther - it is a direct consequence of conservation of angular momentum (areal velocity dA/dt = L/2m is constant).
- Third law (Law of Periods): The square of the time period of a planet is proportional to the cube of the semi-major axis of its orbit. T² ∝ a³, i.e. T²/a³ = constant.
[DIAGRAM: An elliptical orbit with the Sun at one focus; two shaded sectors of equal area near and far from the Sun showing equal areas swept in equal times.]
2. Newton’s Universal Law of Gravitation
Every body in the universe 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
- The force acts along the line joining the two masses and is always attractive.
- It is a vector quantity and obeys Newton’s third law - the two bodies pull each other equally.
- Gravitation is a central force and a conservative force.
3. Universal Gravitational Constant (G)
The constant G is numerically equal to the force of attraction between two unit masses placed unit distance apart.
- Value: G = 6.674 × 10⁻¹¹ N·m²/kg²
- Dimensional formula: [M⁻¹L³T⁻²]
- G is a universal constant - same everywhere, independent of the medium, masses, and nature of the bodies. It was first measured by Henry Cavendish.
Note: Do not confuse G (universal constant) with g (acceleration due to gravity, which varies from place to place).
4. Acceleration Due to Gravity (g)
The acceleration produced in a body due to the gravitational pull of the Earth is called acceleration due to gravity (g). From F = mg and F = GMm/R²:
g = GM/R²
- M = mass of Earth, R = radius of Earth.
- Average value on Earth’s surface: g ≈ 9.8 m/s².
- g is independent of the mass of the falling body - a feather and a stone fall equally fast in vacuum.
Also, g = (4/3)πRρG, where ρ is the mean density of the Earth.
5. Variation of g with Altitude, Depth and Latitude
The value of g is not constant - it changes as you move above, below, or across the surface of the Earth.
With Altitude (height h above surface)
g_h = g(1 − 2h/R), for h ≪ R.
g decreases as you go up. The exact form is g_h = g·R²/(R + h)².
With Depth (depth d below surface)
g_d = g(1 − d/R)
g also decreases as you go down, and becomes zero at the centre of the Earth (d = R).
With Latitude (rotation of Earth)
g’ = g − ω²R cos²λ, where λ is the latitude and ω is the Earth’s angular speed.
- g is maximum at the poles (λ = 90°, cos λ = 0, so no reduction).
- g is minimum at the equator (λ = 0°, maximum reduction).
6. Gravitational Potential Energy
Gravitational potential energy (U) of a body is the work done in bringing it from infinity to a point in the gravitational field.
U = −GMm/r
- It is always negative (taking U = 0 at infinity), because gravity is attractive.
- Near the Earth’s surface, the change in PE for a small height h is the familiar U = mgh.
7. Gravitational Potential
Gravitational potential (V) at a point is the work done in bringing a unit mass from infinity to that point.
V = −GM/r
- SI unit: J/kg. It is a scalar quantity.
- Relation with potential energy: U = mV.
- Relation with field: gravitational field intensity E = −dV/dr.
8. Escape Velocity
Escape velocity (v_e) is the minimum velocity with which a body must be projected so that it escapes the Earth’s gravitational field and never returns.
v_e = √(2GM/R) = √(2gR)
- For Earth, v_e ≈ 11.2 km/s.
- It is independent of the mass and the direction of projection of the body.
- In terms of density: v_e = R√(8πGρ/3).
9. Orbital Velocity of a Satellite
Orbital velocity (v_o) is the velocity needed to keep a satellite revolving in a stable circular orbit. The required centripetal force is provided by gravity.
v_o = √(GM/r) = √[GM/(R + h)]
- For a satellite close to the Earth’s surface (h ≪ R): v_o = √(gR) ≈ 7.9 km/s.
- Important relation: v_e = √2 · v_o - escape velocity is √2 times the orbital velocity of a near-Earth satellite.
Time Period of a Satellite
T = 2π√[(R + h)³/GM] - this is Kepler’s third law in disguise.
10. Satellites - Geostationary and Polar
| Feature | Geostationary Satellite | Polar Satellite |
|---|---|---|
| Orbit plane | Equatorial plane | Passes over the poles (perpendicular to equator) |
| Time period | 24 hours (same as Earth’s rotation) | ≈ 100 minutes (much shorter) |
| Height | ≈ 36,000 km above surface | Low altitude (≈ 500–800 km) |
| Use | Communication, TV, weather | Remote sensing, spying, mapping |
A geostationary satellite appears stationary from the ground because its period matches the Earth’s rotation, so it stays above the same point on the equator.
11. Energy of an Orbiting Satellite
A satellite of mass m in an orbit of radius r has both kinetic and potential energy.
- Kinetic energy: KE = ½mv_o² = +GMm/2r
- Potential energy: PE = −GMm/r
- Total energy: E = KE + PE = −GMm/2r
Key idea: The total energy is negative, which shows the satellite is bound to the Earth. Binding energy = +GMm/2r is the energy needed to remove the satellite to infinity. Also, E = −KE and PE = 2E.
12. Weightlessness
Weightlessness is the state in which a body experiences zero apparent weight. It occurs when the body and its support fall freely under gravity together, so the normal reaction becomes zero.
- An astronaut in an orbiting satellite feels weightless because both the satellite and the astronaut are in free fall toward the Earth with the same acceleration.
- Weightlessness does not mean gravity is absent - g is still significant in orbit; it means there is no reaction force from the floor.
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Weightage in Board & Entrance Exams
| Exam | Typical Weightage | Most-Tested Areas |
|---|---|---|
| CBSE Board (Class 11) | 6–8 marks | Kepler’s laws, variation of g, escape & orbital velocity, satellite energy |
| JEE Main / Advanced | 1–2 questions | Orbital velocity, satellite energy, gravitational potential, variation of g |
| NEET | 1–2 questions | Kepler’s third law, escape velocity, g with altitude/depth, weightlessness |
[TABLE: Question-type split - VSA (1 mark): definitions of G, g, escape velocity; SA (2–3 marks): variation of g, Kepler’s laws, orbital velocity; LA (5 marks): derivation of escape/orbital velocity and energy of a satellite.]
Important Definitions
| Term | Definition |
|---|---|
| Universal law of gravitation | Force between two masses: F = Gm₁m₂/r² - attractive, along the line joining them |
| Gravitational constant (G) | Force between unit masses unit distance apart: G = 6.674 × 10⁻¹¹ N·m²/kg² |
| Acceleration due to gravity (g) | Acceleration of a freely falling body due to Earth’s pull: g = GM/R² |
| Kepler’s third law | Square of period proportional to cube of semi-major axis: T² ∝ a³ |
| Gravitational potential energy | Work done to bring a mass from infinity to a point: U = −GMm/r |
| Gravitational potential | Work done to bring unit mass from infinity to a point: V = −GM/r |
| Escape velocity | Minimum velocity to escape Earth’s field: v_e = √(2gR) ≈ 11.2 km/s |
| Orbital velocity | Velocity for a stable orbit: v_o = √(GM/r); near surface ≈ 7.9 km/s |
| Geostationary satellite | Satellite with a 24-hour period that appears fixed over the equator |
| Weightlessness | State of zero apparent weight during free fall (normal reaction = 0) |
Solved Examples
Example 1
Two masses of 10 kg and 20 kg are placed 1 m apart. Find the gravitational force between them. (G = 6.674 × 10⁻¹¹ N·m²/kg²)
Answer: F = Gm₁m₂/r² = (6.674 × 10⁻¹¹ × 10 × 20)/1² = 1.33 × 10⁻⁸ N.
Example 2
At what height above the Earth’s surface will the value of g become one-fourth of its value on the surface? (R = 6400 km)
Answer: g_h = g·R²/(R + h)². For g_h = g/4, we need (R + h)² = 4R², so R + h = 2R, giving h = R = 6400 km.
Example 3
Calculate the escape velocity from the surface of the Earth. (g = 9.8 m/s², R = 6.4 × 10⁶ m)
Answer: v_e = √(2gR) = √(2 × 9.8 × 6.4 × 10⁶) = √(1.254 × 10⁸) ≈ 11.2 × 10³ m/s = 11.2 km/s.
Example 4
A satellite orbits the Earth close to its surface. Find its orbital velocity. (g = 9.8 m/s², R = 6.4 × 10⁶ m)
Answer: v_o = √(gR) = √(9.8 × 6.4 × 10⁶) = √(6.272 × 10⁷) ≈ 7.92 × 10³ m/s = 7.9 km/s.
Example 5
How does the value of g change at a depth equal to half the radius of the Earth?
Answer: g_d = g(1 − d/R) = g(1 − (R/2)/R) = g(1 − 1/2) = g/2. The value of g is halved.
Example 6
A satellite of mass 200 kg revolves around the Earth in an orbit of radius 7000 km. Find its total energy. (GM = 4 × 10¹⁴ m³/s²)
Answer: E = −GMm/2r = −(4 × 10¹⁴ × 200)/(2 × 7 × 10⁶) = −(8 × 10¹⁶)/(1.4 × 10⁷) ≈ −5.7 × 10⁹ J (negative, so the satellite is bound).
Important Questions for Board Exams
1-Mark Questions (VSA)
- Define gravitational constant G and give its SI unit.
- Why does the value of g become zero at the centre of the Earth?
- What is the relation between escape velocity and orbital velocity of a near-Earth satellite?
- Why is gravitational potential always negative?
- At what place on Earth is the value of g maximum?
2–3-Mark Questions (SA)
- State Kepler’s three laws of planetary motion.
- Derive an expression for the variation of g with altitude (height h above the surface).
- Show that the escape velocity is √2 times the orbital velocity of a satellite near the Earth’s surface.
- What is a geostationary satellite? List two of its uses and one condition it must satisfy.
5-Mark Questions (LA)
- Derive an expression for the orbital velocity and time period of a satellite revolving close to the Earth’s surface.
- Obtain expressions for the kinetic energy, potential energy, and total energy of a satellite in orbit, and explain why the total energy is negative.
- Define escape velocity and derive the expression v_e = √(2GM/R). Hence calculate its value for the Earth.
Quick Revision Points
- Kepler: orbits are ellipses; equal areas in equal times; T² ∝ a³
- Universal law: F = Gm₁m₂/r²; always attractive, central, conservative
- G = 6.674 × 10⁻¹¹ N·m²/kg² (universal); g = GM/R² ≈ 9.8 m/s² (varies)
- g with altitude: g_h = g(1 − 2h/R); with depth: g_d = g(1 − d/R)
- g is max at poles, min at equator; zero at Earth’s centre
- Gravitational PE: U = −GMm/r; potential V = −GM/r (scalar, J/kg)
- Escape velocity: v_e = √(2gR) ≈ 11.2 km/s (independent of mass)
- Orbital velocity: v_o = √(GM/r); near surface ≈ 7.9 km/s; v_e = √2·v_o
- Satellite energy: KE = +GMm/2r, PE = −GMm/r, total E = −GMm/2r
- Geostationary: 24 h period, equatorial, ≈ 36,000 km; polar: low, ≈ 100 min
- Weightlessness: free fall makes normal reaction zero, not zero gravity
Next Chapter: Chapter 8 - Mechanical Properties of Solids
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Gravity is universal
Every lump of matter pulls every other lump along the line joining their centres, and the pull only ever attracts.
r is measured centre-to-centre, never surface-to-surface.
- Same law for an apple and for planets
- Force is always attractive, never repulsive
- Equal and opposite on both masses (Newton’s 3rd law)
Inverse-square law
Force grows with mass but falls off fast with distance, since r is squared in the denominator.
G = 6.67 × 10⁻11 N·m2·kg⁻2 (tiny → gravity matters only for huge masses).
- Double either mass → force doubles
- Halve the distance → force becomes 4×
- Never write G = 9.8 (that’s g, a different quantity)
Acceleration due to gravity g
Set weight mg equal to GMm/R2 and the body’s mass cancels, so every object falls the same way.
g is the pull per kilogram; it depends only on the planet, not on you.
- All bodies fall together because m cancels
- Moon: g ≈ 1.6 m·s⁻2 (about one-sixth)
- Slightly larger at poles than at the equator
How g changes with height & depth
g is largest at the surface and falls off both above and below it, reaching zero at Earth’s centre.
Bracket forms are valid only for small h, d; for large h use g = GM/(R+h)2.
- Height uses 2h/R; depth uses d/R — don’t swap them
- The factor 2 comes from the inverse-square law
- At the centre (d = R), g = 0
Orbital velocity
A satellite is in free fall, moving sideways fast enough that the round Earth curves away beneath it.
Gravity supplies exactly the centripetal force for the circular orbit.
- ≈ 7.9 km/s ≈ 28,000 km/h for a low Earth orbit
- Equivalently v0 = √(gR) since g = GM/R2
- A satellite never stops falling — it keeps missing the ground
Escape velocity
The smallest launch speed to leave a planet forever; from energy, ½mv2 = GMm/R, so m cancels.
Independent of the body’s mass AND of the launch direction/angle.
- Only difference from v0 is the factor 2 inside the root
- vₑ = √2 · v0 ≈ 1.41 × 7.9 ≈ 11.2 km/s
- ‘Just escaping’ = reaching infinity with zero speed (total energy = 0)
Kepler’s three laws
Three exact patterns Kepler found for planets, all later explained by Newton’s inverse-square gravity.
Sun sits at a FOCUS of the ellipse, never at the centre.
- Law of Orbits: ellipse with Sun at one focus
- Law of Areas: equal areas in equal times (angular-momentum conservation)
- Law of Periods: T2 ∝ r3, r = semi-major axis
Using Kepler’s third law
For two bodies round the same Sun, T2/r3 is constant — G and M cancel, so never plug them in.
Fastest at perihelion (closest), slowest at aphelion (farthest).
- 9× farther out → year is √(93) = 27× longer, not 9×
- r is the semi-major axis, not instantaneous distance
- Law of Areas comes from angular momentum, not energy
Gravitational potential & energy
Taking zero at infinity, bringing a mass closer makes the energy negative — the system is bound.
V is potential (J·kg⁻1); potentials add as scalars: V = Σ(−GMᵢ/rᵢ).
- Minus sign = a well, deepest near the mass, rising to 0 at infinity
- You must do positive work to pull the masses apart
- Inside a shell V is constant (= −GM/R) and the field is zero
Change in PE when you rise
Lifting a mass m from the surface to height h costs positive work; mgh is only the small-h shortcut.
For h = R, ΔU = ½mgR, not mgR — g weakens as you rise.
- Result is positive: work done against gravity
- mgh valid only for h much smaller than R
- Uses GM = gR2 to recover the familiar mgh form
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Frequently Asked Questions
It states that every particle attracts every other particle with a force directed along the line joining them, given by F = G m1 m2 / r squared. The force is always attractive and follows an inverse-square law, so halving the distance makes the force four times stronger.
Escape velocity is the minimum speed needed to leave a planet’s gravity forever, ve = square root of (2GM/R), which is about 11.2 km/s for Earth. It equals square root of 2 times the orbital velocity and does not depend on the mass or launch direction of the body.
At a small height h, g decreases by a factor (1 minus 2h/R), and at depth d it decreases by a factor (1 minus d/R). So g is maximum at the surface and becomes zero at the centre of the Earth.
Orbital velocity (about 7.9 km/s) is the speed needed to keep circling a planet just above its surface, while escape velocity (about 11.2 km/s) is the speed needed to break free of its gravity completely. They are related by ve = square root of 2 times vo.
Yes, Gravitation is part of the NEET Physics syllabus and usually contributes one to two questions. Questions are mostly direct and formula-based, focusing on variation of g, escape and orbital velocity, Kepler’s third law T squared proportional to r cubed, and gravitational potential energy.