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Gravitation

Newtons Law of Universal Gravitation

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Newtons law of universal gravitation states that every body in this universe attracts every other body with a force which is directly proportional to the product of their masses and is inversely proportional to the square of the distance between them.

Suppose there are two bodies A and B of masses m1 and m2. Let r be the distance between their centers and F be the force of attraction between them.

 

Newtons Law of Universal Gravitation
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Fm1m2r2

F=Gm1m2r2  (1)

where G is a proportionality constant called the Universal Gravitational Constant.

What is Universal Gravitational Constant “G”?

Universal Gravitational Constant is equal to the force of attraction acting between two bodies, each of unit mass, whose centers are placed a unit distance apart.

Let:

m1=m2=1 and r=1

Then from (1),

F=G1×112=G

G=F

Gravitational constant is a scalar quantity. Its value is the same throughout the universe and is independent of the nature and size of the bodies as well as the nature of the medium between the bodies. The value of G in S.I. is:

G =6.67×1011Nm2kg2

and in the cgs system is:

G =6.67×108dynecm2g2

Dimensional Formula for G

The dimensional formula of G is

=Fr2m1m2 =(MLT2)(L2)M×M

=[M1L3T2]


Remember: Newtons law of universal gravitation holds good for objects lying at very large distances and also at very short distances. It fails when the distance between the objects is less than 10-9m (i.e. of the order of intermolecular distance).


Example: A sphere of mass 40 kg is attracted by a second sphere of mass 15 kg when their centers are 20 cm apart, with a force of 0.1 milligram weight. Calculate the value of gravitational constant.

Solution:

Here, m1=40kg, m2=15kg

r=0.20m

F=0.1 milligram wt. =0.1×106kg wt.

=0.1×106×9.8N

Now, G=Fr2m1m2

G =(0.1×106×9.8)×(0.20)240×15

G =6.53×1011Nm2kg2


Examples of Newtons Law of Universal Gravitation

The rotation of Earth around the Sun is a great example to understand the concept of gravitational attraction. The Earth and the Sun are attracted to each other by the gravitational force of attraction. Due to this gravitational force the Earth would have gone towards the Sun, however the Earth remains in its orbit rotating around the Sun.

This is because of the centrifugal force that acts on the Earth in outward direction. Therefore, the force of gravitation attraction towards the Sun gets balanced by the centrifugal force acting outwards i.e. away from the Sun.

Newtons Law of Universal Gravitation
Image Credit: © Briligence.com

 

The formation of high tides and low tides are also due to the gravitation attraction of the Moon. The gravitational force of attraction between the Earth and the Moon results the ocean water to rise resulting to high tide. When the Moon is on the other side of the Earth there is a low tide on the ocean.

The artificial satellites in their orbit around the Earth is due to the gravitational force of attraction between Earth and satellite. The Earth’s gravity tries to pull the satellite towards  its center, however the centrifugal force of satellite while rotating in a circular orbit around Earth tries to pull outwards i.e. away from the center of the Earth.

Vector Form of Newtons Law of Universal Gravitation

In the vector form the Newtons law of universal gravitation is given by

F=Gm1m2r2

Suppose there are two bodies A and B of masses m1 and m2 placed distance r apart.

r12=unit vector from A to B,

r21=unit vector from B to A

F12 = gravitational force exerted on body A by body B.

F21 = gravitational force exerted on body B by body A.

According to Newtons law of universal gravitation:

F12=Gm1m2r2r^21 (2)

The negative sign indicates that the direction of F12 is opposite to that of r^21. In fact, the negative sign shows that the gravitational force is attractive in nature.

Similarly,

F21=Gm1m2r2r^12 (3)

From (2) and (3), we get:

F12=F21

Important things to know

  1. If the value of G becomes 10 times its present value, then we would be crushed to the floor by Earth’s attraction. If the value of G becomes 110th of its present value, then Earth’s attraction becomes very weak and in that case, we can take jump over a building.
  2. Newton solved the apple-Earth problem by using the shell theorem. According to this theorem, a uniform spherical shell of matter attracts a particle lying outside the shell as if the whole mass of the shell is concentrated at the center of the shell.
  3. No gravitational force acts on a particle due to a spherical shell if the particle is present inside the spherical shell.
  4. The resultant gravitational force on a particle due to other particles is equal to the vector sum of the gravitational forces exerted by individual particles on the given particle. i.e.,

F0=F01+F02+F03+

where F0 is the resultant force on a particle of mass m0, and F01,F02, are the forces on particle of mass m0 due to particles of masses m1,m2,m3,.

Important Characteristics of Gravitational Force

According to Newtons law of universal gravitation, the gravitational force of attraction between two given bodies:

  1. is independent of the nature of the intervening medium.
  2. is independent of the presence or absence of other bodies.
  3. is independent of the nature and size of the bodies, as long as their masses remain the same and the distance between their centers is fixed.
  4. follows action and reaction pair, i.e. the gravitational forces are equal in magnitude and opposite in direction and hence obey Newton’s third law of motion.
  5. The gravitational forces are central forces, as they act along the line joining the centers of the two bodies.
  6. The gravitational forces are conservative forces.

Principle of Superposition of Gravitation

It states that the resultant gravitational force F acting on a particle due to a number of masses is equal to the vector sum of the forces exerted by the individual masses on the given particle, i.e.,

F=F01+F02++F0n =i=1nF0i

where F01,F02,,F0n are the gravitational forces on a particle of mass m0 due to particles of masses m1,m2,m3,,mn, respectively.

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