Class 9

### Topic covered

color{red} ♦ INTRODUCTION
color{red} ♦ GRAVITATION

### INTRODUCTION

As, we have learnt about the motion of objects and force as the cause of motion. We have learnt that a force is needed to change the speed or the direction of motion of an object.

We always observe that an object dropped from a height falls towards the earth. We know that 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. Isaac Newton could grasp that the same force is responsible for all these. This force is called the gravitational force.

In this chapter we shall learn about gravitation and the universal law of gravitation. We shall discuss the motion of objects under the influence of gravitational force on the earth.

We shall study how the weight of a body varies from place to place. We shall also discuss the conditions for objects to float in liquids.

### GRAVITATION

We know that the moon goes around the earth. An object when thrown upwards, reaches a certain height and then falls downwards.

It is said that 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.

Let us try to understand the motion of the moon by recalling activity 8.11.

Activity _____________ 10.1

♦Tie a small stone at one end. Hold the other end of the thread and whirl it round, as shown in Fig. 10.1.
♦ Note the motion of the stone.
♦ Again, note the direction of motion of the stone.

Before the thread is released, the stone moves in a circular path with a certain speed and changes direction at every point. The change in direction involves change in velocity or acceleration.

The force that causes this acceleration and keeps the body moving along the circular path is acting towards the centre. This force is called the centripetal (meaning ‘centre-seeking’) force.

In the absence of this force, the stone flies off along a straight line. This straight line will be a tangent to the circular path.force, the stone flies off along a straight line. This straight line will be a tangent to the circular path.

A straight line that meets the circle at one and only one point is called a tangent to the circle. Straight line ABC is a tangent to the circle at point B.

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.

It is seen that a falling apple is attracted towards the earth. Does the apple attract the earth? If so, we do not see the earth moving towards an apple.

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 [Eq. (9.4)].

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. Extend the same argument for why the earth does not move towards the moon.

In our 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.

From the above facts 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.
Q 3285101067

The mass of the earth is 6 xx 10^(24) kg and that of the moon is 7.4 xx 10^(22) kg. If the distance between the earth and the moon is 3.84 xx 10^5 km, calculate the force exerted by the earth on the moon. G = 6.7 xx 10^(–11) N m^2 kg^(-2).

Class 9 Chapter 10 Example 1
Solution:

The mass of the earth, M = 6 xx 10^(24) kg
The mass of the moon,
m = 7.4 xx 10^(22) kg
The distance between the earth and the moon,
d = 3.84 xx 10^5 km
= 3.84 xx 10^5 xx 1000 m
= 3.84 xx 10^8 m
G = 6.7 xx 10^(–11) N m^2 \ \ kg^(–2)

From Eq. (10.4), the force exerted by the earth on the moon is

 F = G (M xx m)/d^2

 = ( 6.7 xx 10^(-11) N m^2 \ \ kg^2 xx 6 xx 10^(24) kg xx 7.4 xx 10^(22) kg) / ( 3.84 xx 10^8 m)^2

 = 2.01 xx 10^(20) N.

Thus, the force exerted by the earth on the moon is 2.01 xx 10^(20) N.

### 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 as shown in Fig. 10.2. 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. That is,

F ∝ M xx m ......(10.1)

And the force between two objects is inversely proportional to the square of the distance between them, that is,

 F alpha 1/d^2 .....(10.2)

Combining Eqs. (10.1) and (10.2), we get

 F alpha (M xx m)/d^2 .......(10.3)

or, F = G (M xx m)/d^2 ........(10.4)

where G is the constant of proportionality and is called the universal gravitation constant. multiplying crosswise, Eq. (10.4) gives

F xx d^2 = G M xx m

or  G = (F d^2)/(M xx m) .......(10.5)

The SI unit of G can be obtained by substituting the units of force, distance and mass in Eq. (10.5) as N m^2 kg^(–2).

The value of G was found out by Henry Cavendish (1731 – 1810) by using a sensitive balance. The accepted value of G is 6.673 xx 10^(–11) N m^2 kg^(–2).

We know that there exists a force of attraction between any two objects. Compute the value of this force between you and your friend sitting closeby. Conclude how you do not experience this force.

### IMPORTANCE OF THE UNIVERSAL LAW OF GRAVITATION

The universal law of gravitation successfully explained several phenomena which were believed to be unconnected:
(i) the force that binds us to the earth;
(ii) the motion of the moon around the earth;
(iii) the motion of planets around the Sun; and
(iv) the tides due to the moon and the Sun.