Physics Chapter – 4 : Gravitation

Introduction: Gravitation

Centripetal force

We know that an object in circular motion keeps on changing its direction. Due to this, the velocity of the object also changes.

A force called Centripetal Force acts upon the object that keeps it moving in a circular path. It is exerted from the centre of the path.

Without the Centripetal Force objects cannot move in circular paths, they would always travel straight. For Example, The rotation of Moon around the Earth is possible because of the centripetal force exerted by Earth.

Why does Apple fall on Earth from a tree? – Because the earth attracts it towards itself.

Can Apple attract the earth? – Yes. It also attracts the earth as per Newton’s third law (every action has an equal and opposite reaction). But the mass of the earth is much larger than Apple’s mass thus the force applied by Apple appears negligible and Earth never moves towards it.

Newton thus suggested that all objects in this universe attract each other. This force of attraction is called Gravitational Force.

The Universal Law of Gravitation

Newton’s Law of gravitation states that every object in the universe attracts every other object by a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

F∝M∗m
Fα1/d²
F=G Mm/r² where G is the universal gravitation constant.
Value of G = 6.673∗10⁻¹¹Nm²Kg⁻²

Free fall

When an object falls towards the earth due to earth’s gravity and no other force is acting upon it, the object is said to be in free fall state. Free falling objects are not even resisted by the air.

g = 9.8 m/s² is also called the Free-fall Acceleration.

Consider the equations of motion given in different scenarios

When an object at rest falls towards earth – its initial velocity is zero v = gt s = t + (1/2) gt² 2 g s = v²

When an object with some initial velocity (u) falls towards earth – v = u + gt s = ut + (1/2) gt² 2 g s = v²– u²

Acceleration due to gravity

Whenever an object falls towards the Earth there is an acceleration associated with the movement of the object. This acceleration is called acceleration due to gravity.

Denoted by: g
SI Unit: m s^-2
We know that, F= ma
Therefore, F = mg

The following figure demonstrates the mathematical derivation of ‘g’

The force (F) of gravitational attraction on a body of mass m due to earth of mass M and radius R is given by

We know from Newton’s second law of motion that the force is the product of mass and acceleration.
∴ F = ma

But the acceleration due to gravity is represented by the symbol g. Therefore, we can write
F = mg ….. (2)

From the equation (1) and (2), we get

When body is at a distance ‘r’ from the centre of the earth then:

Value of ‘g’ may vary at different parts of the earth

From the equation g = GM/ r² it is clear that the value of ‘g’ depends upon the distance of the object from the earth’s centre.

This is because the shape of the earth is not a perfect sphere. It is rather flattened at poles and bulged out at the equator.

Hence, the value of ‘g’ is greater at the poles and lesser at the equator. However, for our convenience, we take a constant value of ‘g’ throughout.

We can find the value of acceleration due to gravity by the following

Difference between mass and weight:

Mass	Weight
Mass is defined as the quantity of a matter in an object.	The weight of an object is the force by which the gravitational pull of the earth attracts the object.
Mass is the scaler quantity.	Weight is the vector quantity.
The mass of an object is always constant as it depends upon the inertia of the object.	The weight of an object can vary at different locations because of change in gravitational force of the earth.
Masses can never be zero	Weight can be zero at the places there is no gravitational force
Denoted as: m	Denoted as: W F = mg Where m = mass of object a = acceleration due to gravity Similarly, W is force, so W = mg
SI Unit: kg	SI Unit: N

Thrust

The force that acts in the perpendicular direction is called thrust.
It is similar to force applied to an object
It is a vector quantity.

Pressure

The force that acts per unit area of the object is pressure.

It is the thrust per unit area.

Pressure is denoted by ‘P’

P = thrust/ area = force/ area = F/A

SI unit: N/m² or Pa (Pascal)

Buoyancy

Whenever an object is immersed in a liquid, the liquid exerts a buoyant force or upthrust in the opposite direction of the gravitational force. This is also called the Force of Buoyancy.

It depends upon the density of the fluid.

Therefore an object is able to float in water when the gravitational force is less than the buoyant force.

Similarly, an object sinks into the water when the gravitational force is larger than the buoyant force.

Pressure in Fluids

The pressure exerted by a fluid in a container is transmitted undiminished in all directions on the walls of the container.

Archimedes’ Principle

The upward force exerted by a fluid on an object is known as upthrust or buoyant force.

The magnitude of buoyancy depends on the density of the fluid. If the density of an object is less than the fluid, it will float. If the density of the object is greater than the fluid, it will sink.

According to the Archimedes’ principle, when a body is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it.

Relative Density

When density can be expressed in comparison with water’s density it is called Relative Density. It has no unit because it is a ratio of two similar quantities.

Relative density = Density of a substance/ Density of water

Why water is chosen as a reference?

Water is present everywhere on earth so it becomes easier to evaluate the density of a substance in relation to water.

How relative density can be used as a measure to determine in an object will sink or float in water?