Gravitation

What is Gravitation?

• Gravitation is a universal force.
• It acts between objects on the Earth as well as between objects in the Universe.

• The Earth attracts objects towards its centre.
• Therefore, the gravitational force due to the Earth is directed towards the centre of the Earth.

Newton and Gravitation

• Sir Isaac Newton studied the phenomenon of gravitation.
• He wondered why an apple always falls vertically downwards.
• He concluded that the Earth attracts the apple towards itself.
• This attractive force acts towards the centre of the Earth.
• Therefore, the apple falls vertically downwards.

Circular Motion

• An object moving along a circular path continuously changes its direction of motion.
• Therefore, a force must continuously act on the object.
• This force is directed towards the centre of the circular path.

Centripetal Force

• A stone tied to a string can be made to move along a circular path.
• The string pulls the stone towards the centre of the circle.
• If the string is released, the stone moves along the tangent to the circle.

• The word centripetal means centre seeking.

Centripetal Force

Kepler’s Laws

• Johannes Kepler studied the observations of planetary positions.
• He noticed that planets follow certain laws during their motion.
• He stated three laws describing planetary motion.

Kepler’s First Law

• The orbit of a planet is an ellipse.
• The Sun is situated at one of the foci of the ellipse.

Kepler’s Second Law

• A line joining the planet and the Sun can be considered.
• This line sweeps equal areas in equal intervals of time.

Kepler’s Third Law

• The period of revolution of a planet around the Sun is related to its mean distance from the Sun.
• The square of the period of revolution is directly proportional to the cube of the mean distance from the Sun.

Newton’s Universal Law of Gravitation

• Every object in the Universe attracts every other object with a definite force.
• The gravitational force is directly proportional to the product of the masses of the two objects.
• The gravitational force is inversely proportional to the square of the distance between them.

Newton’s Law – Formula

Universal Gravitational Constant – G

• In SI units, G is equal to the gravitational force between two masses of 1 kg kept 1 m apart.
• The SI unit of G is N m² kg⁻².
• The value of G is 6.673 × 10⁻¹¹ N m² kg⁻².
• Henry Cavendish first experimentally measured the value of G.

Inverse Square Law

• A planet moving around the Sun experiences centripetal force.
• The centripetal force is directed towards the Sun.
• For uniform circular motion, the centripetal force is given by F = mv²/r.
• Using Kepler’s third law, the force is found to be inversely proportional to the square of distance.
• Newton identified this force with gravitational force.

Gravitational Force Between Two Objects

• Every object attracts every other object due to gravitational force.
• The gravitational force between ordinary objects is extremely small.
• The textbook example considers two persons of masses 75 kg and 80 kg separated by 1 m.
• Using Newton’s law, the gravitational force between them is 4.002 × 10⁻⁷ N.

Tides

• High and low tides occur because of the gravitational force exerted by the Moon.
• Water directly under the Moon is pulled towards the Moon.
• The water level rises at that place and a high tide occurs.
• At places 90° from the place of high tide, the water level is minimum.
• Therefore, low tides occur at these places.

Earth’s Gravitational Force

• The Earth attracts every object near it towards itself.
• The centre of mass of the Earth is situated at its centre.
• Therefore, the gravitational force due to the Earth is directed towards its centre.
• An object therefore falls vertically downwards towards the centre of the Earth.

Object thrown upwards

• When a stone is thrown vertically upwards, Earth’s gravitational force acts downwards.
• This force reduces the velocity of the stone.
• The velocity becomes zero after the stone reaches a certain height.
• The gravitational force continues to act.
• Therefore, the stone starts moving downwards.

Acceleration Due to Gravity – g

• The Earth exerts gravitational force on objects near it.
• This gravitational force produces acceleration in an object.

• It is denoted by g.
• It is a vector quantity.
• Its direction is towards the centre of the Earth.
• Its SI unit is m/s².

Acceleration Due to Gravity

Variation in the Value of g

A. Along the Surface of Earth

• The Earth is not exactly spherical.
• It bulges at the equator and is flatter at the poles.
• The radius is largest at the equator and smallest at the poles.
• Therefore, g is highest at the poles.
• g is lowest at the equator.

Variation of g with Height and Depth

B. With Height

• When we go above the surface of Earth, the distance from the centre increases.
• Therefore, the value of g decreases.

C. With Depth

• When we go inside the Earth, the value of g decreases.
• At the centre of the Earth, the value of g is zero.

Mass

• The SI unit of mass is kg.
• Mass is a scalar quantity.
• Its value remains the same everywhere.
• Its value does not change when an object is taken to another planet.
• Mass is a measure of inertia.
• Higher the mass, higher is the inertia.

Weight

• The SI unit of weight is Newton (N).
• Weight is a vector quantity.
• Its direction is towards the centre of the Earth.
• Weight changes from place to place because g changes.
• Mass remains constant everywhere.

Mass vs Weight

Gravitational Waves

• Gravitational waves are different from electromagnetic waves.
• Einstein predicted their existence in 1916.
• Gravitational waves are very weak.
• Therefore, they are difficult to detect.
• LIGO is a prominent observatory used to detect gravitational waves.
• Scientists detected these waves exactly 100 years after their prediction.
• Indian scientists contributed significantly to this discovery.

Free Fall

• In free fall, initial velocity is u = 0.
• Acceleration is a = g.
• Velocity increases due to acceleration due to gravity.
• True free fall is possible only in vacuum.

Equations of Free Fall

Object Thrown Vertically Upwards

• Gravitational acceleration acts opposite to the velocity.
• Therefore, acceleration is taken as −g.
• The velocity decreases as the object moves upwards.
• At maximum height, velocity becomes zero.
• The object then starts moving downwards.

Galileo and Free Fall

• The value of g is the same for all objects at a given place on Earth.
• Therefore, two objects of different masses dropped from the same height reach the Earth at the same time if they are falling freely.
• Galileo is said to have performed an experiment around 1590 in Pisa.
• He dropped two spheres of different masses from the leaning tower of Pisa.
• Both spheres reached the ground at the same time.
• In air, a feather and a heavy stone do not reach the ground at the same time because of air resistance and buoyant force.
• In vacuum, a feather and a stone reach the ground at the same time.

Gravitational Potential Energy

• Potential energy is relative.
• For small heights, gravitational potential energy is given by mgh.
• For large heights, the value of g decreases with height.
• At infinite distance from the Earth, gravitational potential energy is taken as zero.
• At smaller distances, gravitational potential energy is negative.

Escape Velocity

• When an object is thrown upwards, its velocity decreases because of Earth’s gravitational force.
• If the initial velocity is sufficiently large, the object can overcome Earth’s gravitational attraction.
• For an object to escape Earth, it must reach an infinite distance from Earth.

Weightlessness in Space

• Space travellers and objects inside a spacecraft appear to float.
• The value of g in space is not zero.
• The spacecraft moves in an orbit around the Earth.
• The gravitational force of the Earth acts on the spacecraft.
• Therefore, the spacecraft is in a state of free fall.
• The spacecraft, travellers and objects inside it have the same free-fall motion.

Important Formulae

Must Remember