What will be the gravitational force between 2 heavenly bodies if the masses of both are tripled keeping the distance between them constant?

Force can be defined as a push or pull that tries to change the state of a body from motion to rest and rest to motion in a straight line. Its S.I. unit is Newton (N) and C.G.S unit is dyne.
Note: 1N = 105 dynes.

Types of force

Generally, there are two types of force.

1. Pulling Force (Attraction Force)
2. Pushing Force (Repulsion Force)

Gravitational force

The force of attraction between any two bodies of the universe is due to their masses is called gravitational force.

Newton’s Universal Law of Gravitation

It states that “The force of gravitation between any two bodies in the universe is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres”.
Let 𝑚1 and 𝑚2 be the mass of two bodies separated by distance d from their centres.

According to law, the gravitational force between these bodies are:

1. directly proportional to the product of their masses, i.e.
   F ∝ m1. m2 ………… (i)
2. inversely proportional to the square of the distance between their centers i.e.
   F ∝ 1/d2 ………… (ii)

Combining equations (i) and (ii),

F\ \alpha\ \frac{m_{1}m_{2}}{d^{2}}\\ F\ = \frac{Gm_{1}m_{2}}{d^{2}}---(iii)

where G is a proportional constant called the universal gravitational constant. The value of G is 6.67 x 10-11 Nm2/kg2.
Equation (iii) gives the gravitational force between two bodies.

Note: Gravitational force depends upon i) mass of the objects ii) distance between two masses from their centre.
Gravitational Force between two small bodies is so small that it is negligible. For an instance, if we take unit masses kept at the unit distance, then the force of attraction between them will be 6.67 x 10-11 N, which is a very small force.

Universal Gravitational Constant

According to Newton’s Law of Gravitation,

F=\frac{Gm_{1}m_{2}}{d^{2}}

If m1 = 1 kg and m2 = 1 kg and d = 1 m, then F = (G x 1 x 1)/12 ⇒ F = G.

So, Universal Gravitational Constant is defined as the gravitational force between any two bodies each of unit mass and separated by unit distance from their centres. Its S.I. unit is Nm2/kg2.

Note: Its value remains constant throughout the universe for all small and big bodies of land, water, air, and space. So, it is called Universal Gravitational Constant.

Properties of Universal Gravitational Constant

• The value of G is independent of the nature of the masses of the body.
• The value of G is unaffected by temperature and pressure
• It is independent of the direction of the gravitational force between two or more masses.
• It is independent of the medium in which masses are placed and the chemical composition of the mass

Consequences of Gravitational Force

• The existence of solar system and constellation is due to the gravitational force between the sun and other heavenly bodies.
• Presence of atmosphere in earth is due to gravitational force between the earth and the atmosphere.
• The tides in the seas and oceans are due to gravitational force between sun and the moon.

Things to remember:

• Tides occur on every new moon day and full moon day when the sun, the moon and the earth lies on the same plane, because of which mutual force of attraction of sun and moon easily affects the flexible water of seas and oceans.
• In liquid molecules, there is less intra molecular force of attraction than in solid, so they are loosely packed as a result the effect of force of gravitation is more in liquid.

Application of Gravitational Force

• It helps to find the gravitational force between two bodies when their masses and distance between the center are known.
• It helps to find distance between two bodies anywhere in the universe if value of gravitational force and their masses are known.

Questions

1. Why is Newton’s Law of gravitation called a universal Law?
2. The force of attraction between any ordinary mass is not remarkable. Justify with an example

Explanation of Newton’s Universal Law of Gravitation

1. Change in the force of attraction when the mass of one body is double keeping the distance between them constant.

Let two bodies of mass 𝑚1 and 𝑚2 are at a distance d each from their centres.
According to Newton’s Universal Law of Gravitation,

F=\frac{Gm_{1}m_{2}}{d^{2}}

When the mass of one object is doubled keeping the distance between them constant, then the new force of gravitation 𝐹’ is calculated as,

F'=\frac{G(2m_{1})m_{2}}{d^{2}} = \frac{2Gm_{1}m_{2}}{d^{2}} = 2F

Therefore, the force of attraction increases by two times when the mass of one body is doubled keeping the distance between their centre constant.

2. Change in the force of attraction when the mass of both bodies is double keeping the distance between them constant.

Let two bodies of mass 𝑚1 and 𝑚2 are at a distance d each from their centres.
According to Newton’s Universal Law of Gravitation,

F=\frac{Gm_{1}m_{2}}{d^{2}}

When the mass of both objects is doubled keeping the distance between them constant, then the new force of gravitation 𝐹′is calculated as:

F'=\frac{G(2m_{1})(2m_{2})}{d^{2}} = \frac{4Gm_{1}m_{2}}{d^{2}} = 4F

Therefore, the force of attraction increases by four times when the mass of both bodies is doubled keeping the distance between them constant.

3. Change in the force of attraction when the mass of both bodies is kept constant and the distance between them is halved.

Let two bodies of mass 𝑚1 and 𝑚2 are at a distance d each from their centres.
According to Newton’s Universal Law of Gravitation,

F=\frac{Gm_{1}m_{2}}{d^{2}}

When the mass of both bodies is kept constant and the distance between them is halved, then the new force of gravitation 𝐹′ is calculated as:

F'=\frac{Gm_{1}m_{2}}{(d/2)^{2}} = \frac{4Gm_{1}m_{2}}{d^{2}} = 4F

Therefore, the force of attraction increases by four times when the mass of both bodies is kept constant and the distance between them is halved.

4. Change in the force of attraction when the mass of both bodies is kept constant and the distance between them is doubled.

Let two bodies of mass 𝑚1 and 𝑚2 are at a distance d each from their centres.
According to Newton’s Universal Law of Gravitation,

F=\frac{Gm_{1}m_{2}}{d^{2}}

When the mass of both bodies is kept constant and the distance between them is doubled, then the new force of gravitation 𝐹′ is calculated as:

F'=\frac{Gm_{1}m_{2}}{(2d)^{2}} = \frac{Gm_{1}m_{2}}{4d^{2}} = F/4

Therefore, the force of attraction decreases by four times when the mass of both bodies is kept constant and the distance between them is halved.