Gravitation

Class 09 Science

A force is needed to change the speed or the direction of motion of an object. An object dropped from a height falls towards the earth. All the planets go around the Sun. The moon goes around the earth.

In all these cases, there must be some force acting on the objects, the planets and on the moon. This force is called the gravitational force.

Gravitation

When Newton was sitting under a tree, an apple fell on him. The fall of the apple made Newton start thinking. He thought that: if the earth can attract an apple, can it not attract the moon? Is the force the same in both cases?

He conjectured that the same type of force is responsible in both the cases. He argued that at each point of its orbit, the moon falls towards the earth, instead of going off in a straight line. So, it must be attracted by the earth. But we do not really see the moon falling towards the earth.

The motion of the moon around the earth is due to the centripetal force. The centripetal force is provided by the force of attraction of the earth. If there were no such force, the moon would pursue a uniform straight line motion.

According to the third law of motion, the apple does attract the earth. But according to the second law of motion, for a given force, acceleration is inversely proportional to the mass of an object. The mass of an apple is negligibly small compared to that of the earth. So, we do not see the earth moving towards the apple.

In the solar system, all the planets go around the Sun. By arguing the same way, we can say that there exists a force between the Sun and the planets. Newton concluded that not only does the earth attract an apple and the moon, but all objects in the universe attract each other. This force of attraction between objects is called the gravitational force.

Universal Law of Gravitation

Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force is along the line joining the centres of two objects.

Let two objects A and B of masses M and m lie at a distance d from each other. Let the force of attraction between two objects be F.

According to the universal law of gravitation, the force between two objects is directly proportional to the product of their masses.

$$ F \propto M \times m $$

The force between two objects is inversely proportional to the square of the distance between them.

$$ F \propto \frac{1}{d^2} $$

Combining both equations,

$$ F \propto \frac{M \times m}{d^2} $$

$$ F = G\, \frac{M \times m}{d^2} $$

where G is the constant of proportionality and is called the universal gravitation constant.

$$ G = \frac{F \cdot d^2}{M \times m} $$

The SI unit of G is N m² kg^–2.

G = 6.673 x 10^–11 N m² kg^–2

Importance of Universal Law of Gravitation

The universal law of gravitation successfully explained several phenomena which were believed to be unconnected:

1. the force that binds us to the earth
2. the motion of the moon around the earth
3. the motion of planets around the Sun
4. tides due to the moon and the Sun

Free Fall

The earth attracts objects towards it. This is due to the gravitational force. Whenever objects fall towards the earth under this force alone, we say that the objects are in free fall.

While falling, there is no change in the direction of motion of the objects. But due to the earth’s attraction, there will be a change in the magnitude of the velocity. Whenever an object falls towards the earth, an acceleration is involved. This acceleration is due to the earth’s gravitational force. Therefore, this acceleration is called the acceleration due to to gravity. It is denoted by g. The unit of g is the same as that of acceleration: m s^–2.

The magnitude of the gravitational force F will be equal to the product of mass and acceleration due to the gravitational force.

$$ F = m g $$

$$ mg = G\, \frac{M \times m}{d^2} $$

$$ g = G\, \frac{M}{d^2} $$

where M is the mass of the earth, and d is the distance between the object and the earth.

If the object is on or near the surface of the earth, the distance d will be equal to R, the radius of the earth. Thus, for objects on or near the surface of the earth,

$$ g = G\, \frac{M}{R^2} $$

The earth is not a perfect sphere. As the radius of the earth increases from the poles to the equator, the value of g becomes greater at the poles than at the equator.

Value of g

Universal gravitational constant, G = 6.7 × 10^–11 N m² kg^-2

Mass of the earth, M = 6 × 10²⁴ kg

Radius of the earth, R = 6.4 × 10⁶ m

g = 9.8 m s^–2

Gravitational Motion

An object experiences acceleration during free fall. This acceleration experienced by an object is independent of its mass. This means that all objects fall at the same rate.

As g is constant near the earth, all the equations for the uniformly accelerated motion of objects become valid with acceleration a replaced by g.

Mass

Mass of an object is the measure of its inertia. Greater the mass, the greater is the inertia. It remains the same whether the object is on the earth, the moon or even in outer space. Thus, the mass of an object is constant and does not change from place to place.

Weight

The earth attracts every object with a certain force and this force depends on the mass (m) of the object and the acceleration due to the gravity (g). The weight of an object is the force with which it is attracted towards the earth.

F = m × g

The force of attraction of the earth on an object is known as the weight of the object. It is denoted by W.

W = m × g

As the weight of an object is the force with which it is attracted towards the earth, the SI unit of weight is the same as that of force, that is, newton (N). The weight is a force acting vertically downwards - it has both magnitude and direction.

The value of g is constant at a given place. Therefore, at a given place, the weight of an object is directly proportional to the mass, say m, of the object.

W ∝ m

It is due to this reason that at a given place, we can use the weight of an object as a measure of its mass. The mass of an object remains the same everywhere, that is, on the earth and on any planet whereas its weight depends on its location because g depends on location.

Thrust and Pressure

Weight is the force acting vertically downwards. The force acting on an object perpendicular to the surface is called thrust.

The thrust on unit area is called pressure.

$$ \text{Pressure} = \frac{\text{Thrust}}{\text{Area}} $$

SI unit of pressure as N/m² or N m^–2. The SI unit of pressure is called pascal, denoted as Pa.

The same force acting on a smaller area exerts a larger pressure, and a smaller pressure on a larger area. This is the reason why a nail has a pointed tip, knives have sharp edges and buildings have wide foundations.

Pressure in Fluids

All liquids and gases are fluids. A solid exerts pressure on a surface due to its weight. Similarly, fluids have weight, and they also exert pressure on the base and walls of the container in which they are enclosed. Pressure exerted in any confined mass of fluid is transmitted undiminished in all directions.

Buoyancy

Take an empty plastic bottle. Close the mouth of the bottle with an airtight stopper. Put it in a bucket filled with water. You see that the bottle floats.

The force due to the gravitational attraction of the earth acts on the bottle in the downward direction. So the bottle is pulled downwards. But the water exerts an upward force on the bottle. Thus, the bottle is pushed upwards.

Weight of an object is the force due to gravitational attraction of the earth. When the bottle is immersed, the upward force exerted by the water on the bottle is greater than its weight. Therefore it rises.

To keep the bottle completely immersed, the upward force on the bottle due to water must be balanced. This can be achieved by an externally applied force acting downwards. This force must at least be equal to the difference between the upward force and the weight of the bottle.

The upward force exerted by the water on the bottle is known as upthrust or buoyant force. All objects experience a force of buoyancy when they are immersed in a fluid. The magnitude of this buoyant force depends on the density of the fluid.

Why Objects Float or Sink

Objects of density less than that of a liquid float on the liquid. The objects of density greater than that of a liquid sink in the liquid.

Archimedes’ Principle

The upward force exerted by water is known as the force of buoyancy.

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.

Archimedes’ principle has many applications. It is used in designing ships and submarines. Lactometers, which are used to determine the purity of a sample of milk and hydrometers used for determining density of liquids, are based on this principle.