Gravitation

We have learnt about the motion of objects and force as the cause of motion in the previous chapters. 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.

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 t he 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

 

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.

 

 

Fig.1

 

 Let two objects A and B of masses M and m lie at a distance d from each other as shown in Fig. 1.. 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*m           ---------- (1)                                                    

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

                                     F α 1 ⁄ d²       ---------- (2)

                                                           

Combining Eqs. (1) and (2), we get

                     

                                F α M x m ⁄ d²      ---------- (3)

                    or, F = G( (M x m) ⁄ d²)    ---------- (4)                                    

 

                          

where G is the constant of proportionality and is called the universal gravitation constant. By multiplying crosswise, Eq. (4) gives

                              F × d ² = G M × m

    or                       G  = F d²/ M m        ---------- (5)           

                                                     

The SI unit of G can be obtained by substituting the units of force, distance and mass in Eq. (5) as N m² 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 × 10^–11 N m² 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:

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; and

4.   the tides due to the moon and the Sun.

 

Free Fall

We have learnt that 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. Is there any change in the velocity of falling objects? 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. Any change in velocity involves acceleration. 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 the gravitational force of the earth (or acceleration due to gravity). It is denoted by g. The unit of g is the same as that of acceleration, that is, m s^–2.

We know from the second law of motion that  force is the product of mass and acceleration. Let the mass of the stone in be m. We already know that there is acceleration involved in falling objects due to the gravitational force and is denoted by g. Therefore the magnitude of the gravitational force F will be equal to the product of mass and acceleration due to the gravitational force, that is,

F = m g --------(6)

From Eqs. (4) and (6) we have
mg =G((M × m) ⁄ d²)
or g = G(M ⁄ d²) --------(7)

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

Let an object be on or near the surface of the earth. The distance d in Eq. (7) will be equal to R, the radius of the earth. Thus, for objects on or near the surface of the earth,

mg =G((M × m)  ⁄  R2) --------(8)
g = G(M ⁄ R2)                --------(9)

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. For most calculations, we can take g to be more or less constant on or near the earth. But for objects far from the earth, the acceleration due to gravitational force of earth is given by Eq. (7).

To calculate the value of  g:

To calculate the value of g, we should put the values of G, M and R in Eq. (9), namely, universal gravitational constant,

G = 6.7 × 10^–11 N m² kg^-2, mass of the earth,
M = 6 × 10²⁴ kg, and radius of the earth,
R = 6.4 × 10⁶ m.

 g=G M/R²

  6.7 10^-11 N m² kg^-2 6 10²⁴ kg / (6.4 10⁶ m)²  = 9.8 m s^–2.

Thus, the value of acceleration due to gravity of the earth, g = 9.8 m s^–2.

 

Mass

We have learnt in the previous chapter that the mass of an object is the measure of its inertia. We have also learnt that 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

We know that 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.

   We know that:

                           F=m*a     --------(10)                                         

That is

                           F =m*g    --------(11)                   

                

The force of attraction of the earth on an object is known as the weight of the object. It is denoted by W. Substituting the same in Eq. (11), we have

                                w=m*g   --------(12)                                   

 

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.

We have learnt that 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, that is, 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.

 

Weight of  an object on the moon

We have learnt that the weight of an object on the earth is the force with which the earth attracts the object. In the same way, the
weight of an object on the moon is the force with which the moon attracts that object. The mass of the moon is less than that of the earth. Due to this the moon exerts lesser force of attraction on objects.

Let the mass of an object be m. Let its weight on the moon be W_m. Let the mass of the moon be Mm and its radius be R_m.

By applying the universal law of gravitation, the weight of the object on the moon will be

                W_m = G((M_m × m) ⁄ R²_m)         --------(13)                                                            

Let the weight of the same object on the earth be W_e. The mass of the earth is M and its radius is R.

Table-1:

Celestial  body	Mass(kg)	Radium(m)
Earth	5.98  1024	6.37  106
Moon	7.36  1022	1.74   106

                        

From Eqs. (9) and (12) we have,

            W_e=G ((M × m) ⁄ R²)  --------(14)          

Substituting the values from Table 1 in Eqs. (13) and (14), we get

           W_m = G ((7.36 × 10²²kg × m) ⁄ (1.74 × 10⁶m)²)
           W_m= 2.431×10¹⁰ G × m                                      --------(15-a)
and    W_e= 1.474 × 10¹¹ G × m                                      --------(15-b)

Dividing Eq. (15-a) by Eq. (15-b), we get

W_m ⁄ W_e = (2.431 × 10¹⁰) ⁄ (1.474 × 10¹¹)
W_m ⁄ W_e = 0.165 ≈ 1 ⁄ 6                                                  --------(16)      

Wieght of the Object on Moon  ⁄  Weight of the Object on Earth = 1 ⁄ 6

Weight of the object on the moon = (1/6) × its weight on the earth.

 

Summary :

• The law of gravitation states that the force of attraction between any two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. The law applies to objects anywhere in the universe. Such a law is said to be universal.

• Gravitation is a weak force unless large masses are involved.

• Force of gravitation due to the earth is called gravity.

• The force of gravity decreases with altitude. It also varies on the surface of the earth, decreasing from poles to the equator.

• The weight of a body is the force with which the earth attracts it.

• The weight is equal to the product of mass and acceleration due to gravity.

• The weight may vary from place to place but the mass stays constant.