Gravitation Class 10 Notes, Formulas, Questions | SSC

This chapter explains how the gravitational force, first understood by Newton, acts between every pair of objects in the universe. It starts with circular motion and centripetal force, moves through Kepler's three laws of planetary motion and shows how they led Newton to the inverse square law. It then studies the acceleration due to gravity, how g changes with place, height and depth, the difference between mass and weight, free fall and the equations for it, gravitational potential energy and escape velocity. It closes with weightlessness in space and gravitational waves.

Summary

A force is needed to change the speed or the direction of motion of an object. When a stone tied to a string is whirled in a circle, the hand pulls it towards the centre; if the string is released, the stone flies off along the tangent, which is the direction of its velocity at that moment. A force that acts on a body moving in a circle and is directed towards the centre is called the centripetal (centre seeking) force. In the same way the earth must pull the moon, and the Sun must pull the planets, to keep them in their orbits.

Johannes Kepler studied the planetary observations of Tycho Brahe and stated three laws. First law: the orbit of a planet is an ellipse with the Sun at one focus. Second law: the line joining the planet and the Sun sweeps equal areas in equal intervals of time, so a planet moves faster when it is nearer the Sun. Third law: the square of the period of revolution of a planet is directly proportional to the cube of its mean distance from the Sun, that is T squared divided by r cubed is a constant K. Kepler found these laws from observations but could not explain why planets obey them.

Newton used the centripetal force formula F = mv squared/r for a planet in a circular orbit. Writing the speed as 2 pi r/T and using Kepler's third law, he showed that the force on the planet is inversely proportional to the square of its distance from the Sun. Identifying this force with gravity, he gave the universal law of gravitation: every object in the universe attracts every other object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. G is the universal gravitational constant, first measured by Henry Cavendish; its value is 6.673 x 10^-11 N m^2 kg^-2. For spherical bodies the distance is measured between the centres; for irregular bodies, between their centres of mass. Gravity is much weaker than the other forces in nature, yet it controls the universe because planets and stars have huge masses. The moon's pull also causes high and low tides on the earth.

The earth's gravitational force acts towards its centre, so objects fall vertically downwards, and a stone thrown up slows down, stops and falls back. The moon and artificial satellites do not fall on the earth because of their velocity along the orbit. The acceleration produced in a body by the earth's gravitational force is called the acceleration due to gravity, g. From mg = GMm/r squared we get g = GM/r squared, and on the surface g = GM/R squared, which is about 9.8 m/s^2. It depends only on the mass and radius of the earth, not on the falling object.

The value of g changes. Along the surface it is highest at the poles (9.832 m/s^2) and lowest at the equator (9.78 m/s^2) because the earth bulges at the equator. It decreases with height, slightly for an aeroplane but a lot for satellites (about 0.225 m/s^2 at the height of a communication satellite). It also decreases with depth, because the mass of the earth that pulls the object becomes smaller, and it becomes zero at the centre. On the moon g is about one sixth of that on the earth.

Mass is the amount of matter in an object; it is a scalar, its SI unit is kg, it is the measure of inertia and it is the same everywhere. Weight is the force with which the earth attracts the object, W = mg; it is a vector directed towards the centre of the earth, its SI unit is newton and it changes from place to place because g changes. In daily language we say weight in kg but we actually mean mass.

When a body moves under the influence of gravity alone, it is in free fall. For free fall from rest u = 0 and a = g, so v = gt, s = half g t squared and v squared = 2gs. For upward motion the acceleration is taken as -g. All bodies dropped from the same height fall together in the absence of air, as Galileo is said to have shown at Pisa around 1590; a feather falls slowly in air because of air friction and buoyancy. The moon and satellites are also in free fall, and this is why astronauts and objects in a spacecraft appear weightless, even though g at the space station is only about 11% less than on the surface.

Potential energy of a body at height h is mgh only when h is small. Taking the potential energy to be zero at infinity, the potential energy at height h is -GMm/(R+h). A ball thrown up reaches a maximum height u squared/2g, and if its initial velocity is large enough it never returns. This minimum velocity is the escape velocity; using conservation of energy, v_esc = square root of 2GM/R, which is 11.2 km/s for the earth. Spacecraft sent to other planets must exceed it. Gravitational waves, predicted by Einstein in 1916, are waves in the fabric of space-time; they were detected a hundred years later by instruments such as LIGO, with contributions from Indian scientists.

Mind map

How Newton arrived at the inverse square law

Important definitions

Formulas

Key points to remember

• A whirled stone released from the string moves along the tangent to the circle.
• Kepler gave his laws from the observations of Tycho Brahe; Newton later derived them from his theory of gravity.
• A planet moves faster when it is closer to the Sun (second law).
• Doubling one mass doubles the gravitational force; doubling the distance makes the force one fourth.
• G is a universal constant; g changes from place to place.
• g on the earth's surface is about 9.8 m/s^2: 9.832 m/s^2 at the poles and 9.78 m/s^2 at the equator.
• g decreases with height and with depth and is zero at the centre of the earth.
• g on the moon is about one sixth of g on the earth, so weight on the moon is about one sixth.
• Mass is the same everywhere; weight changes with g.
• Objects in free fall fall together regardless of mass; true free fall is possible only in vacuum.
• Astronauts feel weightless because they are in free fall, not because g is zero.
• Escape velocity of the earth is 11.2 km/s and of the moon about 2.37 km/s.

Solved examples

Example 1. Two persons of masses 75 kg and 80 kg sit 1 m apart. Find the gravitational force between them. (G = 6.67 x 10^-11 N m^2/kg^2)

Given \( m_1 = 75 \) kg, \( m_2 = 80 \) kg, \( d = 1 \) m. \( F = \dfrac{G m_1 m_2}{d^2} = \dfrac{6.67\times10^{-11}\times75\times80}{1^2} = 4.002\times10^{-7} \) N. The force is very small, which is why we do not notice it.

Example 2. Calculate g on the earth's surface. Mass of earth = 6 x 10^24 kg, radius = 6.4 x 10^6 m.

\( g = \dfrac{GM}{R^2} = \dfrac{6.67\times10^{-11}\times6\times10^{24}}{(6.4\times10^{6})^2} = \dfrac{4.002\times10^{14}}{4.096\times10^{13}} \approx 9.77 \) m/s^2.

Example 3. A person weighs 750 N on the earth. The moon's mass is 1/81 and its radius 1/3.7 of the earth's. Find his weight on the moon.

Weight on earth \( mg_E = \dfrac{GM_E m}{R_E^2} = 750 \). Weight on moon \( = \dfrac{GM_M m}{R_M^2} = 750\times\dfrac{M_M}{M_E}\times\dfrac{R_E^2}{R_M^2} = 750\times\dfrac{1}{81}\times(3.7)^2 \approx 126.8 \) N, nearly one sixth of the weight on earth.

Example 4. An iron ball of mass 3 kg falls freely from 125 m. Taking g = 10 m/s^2, find (i) time to reach the ground, (ii) velocity on reaching the ground, (iii) its height at half that time.

(i) \( s = ut + \tfrac12 at^2 \Rightarrow 125 = 0 + 5t^2 \Rightarrow t^2 = 25,\ t = 5 \) s. (ii) \( v = u + at = 0 + 10\times5 = 50 \) m/s. (iii) Half time = 2.5 s; distance fallen \( = \tfrac12\times10\times(2.5)^2 = 31.25 \) m, so height \( = 125 - 31.25 = 93.75 \) m. The mass of the ball does not matter.

Example 5. A tennis ball thrown up reaches 4.05 m. Find its initial velocity and total time in air (g = 10 m/s^2).

Upward: v = 0, a = -10 m/s^2, s = 4.05 m. \( v^2 = u^2 + 2as \Rightarrow 0 = u^2 - 81 \Rightarrow u = 9 \) m/s. Downward: \( 4.05 = \tfrac12\times10\times t^2 \Rightarrow t^2 = 0.81,\ t = 0.9 \) s. Time up equals time down, so total time = 1.8 s.

Example 6. Find the escape velocity on the moon. Mass = 7.34 x 10^22 kg, radius = 1.74 x 10^6 m.

\( v_{esc} = \sqrt{\dfrac{2GM}{R}} = \sqrt{\dfrac{2\times6.67\times10^{-11}\times7.34\times10^{22}}{1.74\times10^{6}}} \approx \sqrt{5.63\times10^{6}} \approx 2.37\times10^{3} \) m/s = 2.37 km/s.

Example 7. An object takes 5 s to fall 5 m from rest on a planet. Find g on that planet.

\( s = \tfrac12 g t^2 \Rightarrow 5 = \tfrac12\times g\times25 \Rightarrow g = \dfrac{10}{25} = 0.4 \) m/s^2.

Example 8. The period of a planet at distance R from a star is T. Find its period at distance 2R.

By Kepler's third law \( \dfrac{T^2}{R^3} = \dfrac{T'^2}{(2R)^3} \Rightarrow T'^2 = 8T^2 \Rightarrow T' = \sqrt{8}\,T = 2\sqrt{2}\,T \).

Important questions and answers

Q1. State Kepler's three laws of planetary motion.

First: the orbit of a planet is an ellipse with the Sun at one focus. Second: the line joining the planet and the Sun sweeps equal areas in equal intervals of time. Third: the square of the period of revolution of a planet is directly proportional to the cube of its mean distance from the Sun (T squared/r cubed = constant).

Q2. State Newton's universal law of gravitation and give the SI unit of G.

Every object in the universe attracts every other object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them: F = G m1 m2 / d squared. From G = F d squared/(m1 m2), the SI unit of G is N m^2 kg^-2; its value is 6.673 x 10^-11 N m^2 kg^-2.

Q3. Distinguish between mass and weight.

Mass is the amount of matter in a body, weight is the force with which the earth attracts it. Mass is scalar, weight is vector (towards the centre of the earth). SI unit of mass is kg, of weight is newton. Mass is the same everywhere and measures inertia; weight changes from place to place because g changes (W = mg).

Q4. Give reason: The value of g is maximum at the poles and minimum at the equator.

The earth is not perfectly spherical; due to its rotation it bulges at the equator and is flat at the poles. Since g = GM/R squared, the larger radius at the equator gives a smaller g (9.78 m/s^2), and the smaller radius at the poles gives a larger g (9.832 m/s^2).

Q5. Explain why the value of g is zero at the centre of the earth.

As we go inside the earth, the part of the earth's mass that attracts the object decreases. At the centre the earth's mass surrounds the object equally on all sides, so the pulls cancel and the effective attracting mass is zero. Hence g becomes zero at the centre.

Q6. Give reason: Astronauts in a spacecraft orbiting the earth feel weightless.

At the height of the space station g is only about 11% less than at the surface, so it is not zero. The spacecraft, astronauts and objects inside are all in free fall because gravity is the only force on them, and the velocity of free fall does not depend on the object. So everything falls together and an object released by an astronaut stays at rest relative to her, appearing weightless.

Q7. What is free fall? Is true free fall possible in air?

Motion of an object under the influence of gravity alone is free fall. In air, friction due to air and buoyant force also act on the object, so true free fall is possible only in vacuum.

Q8. Why does a feather fall more slowly than a stone in air, though g is the same for both?

The feather experiences buoyant force and air friction comparable to its small weight, so it floats down slowly. For the stone these forces are much smaller than its weight. In vacuum both reach the ground together.

Q9. What is escape velocity? Derive its expression.

It is the minimum initial velocity a body needs to escape the earth's gravity. On the surface: KE = half m v_esc squared, PE = -GMm/R. At infinity KE = 0 and PE = 0. By conservation of energy, half m v_esc squared - GMm/R = 0, so v_esc = square root of (2GM/R) = square root of (2gR) = 11.2 km/s for the earth.

Q10. Show that the time taken by a stone to go up equals the time taken to come down.

Going up with initial velocity u: v = 0 = u - g t1, so t1 = u/g, and height h = u squared/2g. Coming down from rest through h: h = half g t2 squared, so t2 = square root of (2h/g) = square root of (u squared/g squared) = u/g. Hence t1 = t2.

Q11. If g suddenly doubled, why would it be twice as hard to pull a heavy object along the floor?

The weight mg doubles, so the normal force from the floor doubles. Friction is proportional to this normal force, so the frictional force opposing the pull also doubles and twice the force is needed.

Q12. Why does the apple fall towards the earth but the earth does not visibly move towards the apple?

Both exert equal forces on each other, but acceleration = force/mass. The earth's mass is enormous, so its acceleration is negligible, while the apple's small mass gives it a large acceleration.

Exam tips

• Write Kepler's laws in exact wording and add the relation T squared/r cubed = K for the third law.
• In numericals, always write Given, formula, substitution and final answer with unit.
• Use g = 9.8 m/s^2 unless the question gives g = 10 m/s^2.
• Remember G is constant everywhere, while g varies with place, height and depth.
• For ratio problems (weight on moon, period at 2R), divide the two formulas so G and unknown masses cancel.
• Keep escape velocity values ready: earth 11.2 km/s, moon about 2.37 km/s.

FAQs

What is the difference between G and g?

G is the universal gravitational constant (6.67 x 10^-11 N m^2/kg^2), same throughout the universe. g is the acceleration due to gravity of a particular planet (about 9.8 m/s^2 on earth), which varies with place.

Why do satellites not fall on the earth?

They have a velocity along their orbit. The earth's pull only keeps changing their direction, acting as the centripetal force, so they keep moving around the earth instead of falling.

Does the mass of a body change on the moon?

No. Mass stays the same; only the weight becomes about one sixth because g on the moon is about one sixth of that on the earth.

Which instrument detected gravitational waves?

LIGO (Laser Interferometric Gravitational Wave Observatory) detected them, exactly a hundred years after Einstein predicted them in 1916.

These notes were prepared with AI help from the Balbharati Science and Technology Part 1 textbook (2026-27) and checked against it.