Newton’s Law of Universal Gravitation states that:

Every particle attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Consider two particles of masses m1 and m2 separated by a distance r. Each will exert a force F on the other, given by

where F : gravitational force between the two particles. m1, m2: masses of the two particles.

r: distance between the two particles.

G: constant of universal gravitation.

m1

m2

F

F

r

m1

m2

F

F

r

Figure 1

The two forces form an action-reaction pair and have the following characteristics, * are equal in magnitude,

* are opposite in direction,

* act on different bodies

* are of the same type (gravitational force).

G is a universal constant called the gravitational constant (or constant of universal gravitation), which has been measured experimentally to be : G = 6.67 x 10-11 N m2 kg-2.

Important points to note about Newton’s Law of Gravitation

1.Newton’s Law of Gravitation is a universal law. It applies everywhere in the universe.

2.Attractive Nature of gravitational force: Note that the particles in this case are always attracted to each other.

3.The gravitational force is a field force that always exists between two point masses regardless of the medium that separates them. In fact, it would still exist if there were no medium between them.

4.Inverse-square law:

Newton’s Law of Gravitation is an example of an inverse-square law i.e. the force is inversely proportional to the square of the separation of the particles.

F

5.Even though the law is stated for point masses, the law could also be applied for the attraction exerted on an external object by a spherical object with radial symmetry, e.g.

(a) sphere of uniform density;

(b) spherical shell of uniform density;

(c) sphere composed of uniform concentric shells.

The object will behave as if its whole mass was concentrated at its centre.

r would represent the distance between the centres of mass of the two bodies.

Example 1:Does the Earth fall towards the apple?

Suppose an apple drops from a tree and move towards the ground. Estimate the acceleration of the Earth towards the apple. (You may assume that mass of Earth to be ~ 1024 kg.)

Solution:

To solve this problem, we need to first determine why the Earth falls towards the apple. Because both the apple and Earth have mass,

* The apple is attracted towards the Earth, i.e. Earth exerts a gravitational force on the apple. * Similarly, by Newton’s 3rd Law, the apple will exert a gravitational force on the apple.

Assuming that no other forces act on the Earth, then the gravitational force acting on the Earth will cause the Earth to accelerate. (Newton’s 2nd Law).

By Newton’s 3rd Law, the gravitational force acting on the Earth is equal in magnitude to the gravitational force acting on the apple, i.e.

FEarth= Fapple

= Weight of the apple

= mg = (0.1 kg) (9.81 m s-2) = 0.981 N

By Newton’s 2nd Law,

The acceleration of Earth, a =

=

~ (1 N)/(1024 kg)

~ 10-24 m s-2

What is the physical significance of this calculation?

Although both the apple and the Earth experience the same magnitude of force, the mass of Earth is much larger. Hence, acceleration a is so small that the motion of the Earth is negligible.

m2

m2

Example 2: Finding Resultant Gravitational Force (Serway and Vuille. p209. Example 7.10)

Three 0.300 kg billiard balls are placed on a table at the corners of a right triangle. 0.400 m

0.400 m

0.500 m

0.500 m

Find the resultant gravitational force on the cue ball (designated as m1), resulting from the other two balls. Remember to state the direction of the resultant gravitational force as well.

0.300 m

0.300...