__What is Gravitational Force?__

In universe, all bodies attract to other bodies with a *force* of attraction called * force of gravitation or gravitational force. *This force of attraction depends upon the mass of interacting bodies and the distance between them.

*Therefore, gravitational force is the force of attraction between any two bodies in the universe by virtue of their masses.*

__Gravity__

When earth is one of the bodies between two interacting bodies resulting in gravitational force of attraction between them, then the force of attraction is called **gravity.**

*Therefore, gravity is the force of attraction by which earth attracts any object lying on its surface or near to its surface.*

__Features of Gravitational Force__

Important features of gravitational forces are –

- It is a force between two masses and doesn’t depend upon the medium.
- Mutual gravitational forces between two bodies are equal and opposite, hence obeys
*Newton’s third law of motion.* - Law of gravitation strictly holds for point masses.
- Gravitational force is a central force which acts along the line joining
*centres of mass*of bodies. - The gravitational force possess spherical symmetry.
- Gravitational force is a
*conservative force*(A conservative force is a force with the property that total*work done*in moving a particle between two points is not dependent on the path of movement of that particle. It depends only on the start point and end point. Therefore, total work done by a conservative force in moving a particle in a circle will comes to zero.) - The gravitational force between two bodies is independent of presence of other bodies.

__Newton’s Laws of Gravitation__

Newton stated three laws for gravitational force of attraction between two interacting bodies which are commonly called as ** universal laws of gravitation**.

These laws are stated as –

### First Law of Gravitation

Newton’s first law of gravitation states that –

*Gravitational force of attraction between two bodies is directly proportional to the product of their masses.*

Consider about two bodies of masses ( m_1 ) \ \text {and} \ ( m_2 ) are kept at a distance ( r ) apart as shown in figure.

Therefore, \quad F \propto \left ( { m_1 }{ m_2 } \right ) ………. (1)

### Second Law of Gravitation

Newton’s second law of gravitation states that –

*Gravitational force of attraction between two bodies is inversely proportional to the square of the distance between them.*

Therefore, \quad F \propto \left ( \frac { 1 }{ r ^2 } \right ) ……… (2)

### Third Law of Gravitation

Newton’s third law of gravitation states that –

*The force of gravitation acts along the line joining the centre of masses of two interacting bodies.*

By combining the relations (1) and (2), we get –

F \propto \left ( \frac { m_1 m _2 }{ r^2 } \right )

Or, \quad F = G \left ( \frac {m_1 m_2 }{ r ^2 } \right )

Where ( G ) is a constant of proportionality which is popularly known as *universal gravitational constant.*

__Universal Gravitational Constant__

According to Newton’s laws of gravitation we have –

F = G \left ( \frac { m_1 m_2 }{ r^2 } \right )

If \quad ( m_1 = 1 ), \quad ( m_2 = 1 ) and \quad ( r = 1 )

Then, Universal Gravitational Constant \quad G = F

*Therefore, universal gravitational constant is defined as the force of attraction between two bodies of unit masses placed at unit distance apart.*

__Superposition of Gravitational Force__

Principal of superposition of gravitational forces states that –

*The gravitational force between two masses acts independently and not influenced by the presence of other mass bodies.*

Consider about a particle of mass ( m_0 ) surrounded by three other particles of masses ( m_1 ), \ ( m_2 ) and ( m_3 ) as shown in figure.

According to the laws of gravitation –

- First force on mass ( m_0 ) exerted by the mass ( m_1 ) will be – \quad F_1 = G \left ( \frac { m_1 m_0 }{ r_1^2 } \right )
- Second force on mass ( m_0 ) exerted by mass ( m_2 ) will be – \quad F_2 = G \left ( \frac { m_2 m_0 }{ r_2^2 } \right )
- Third force on mass ( m_0 ) exerted by mass ( m_3 ) will be – \quad F_3 = G \left ( \frac { m_3 m_0 }{ r_3^2 } \right )

Therefore, *resultant force* ( F_R ) , acting on mass ( m_0 ) will be the vector sum of the forces ( F_1 ), \ ( F_2 ), \ \text {and} \ ( F_3 )

Or, \quad F_R = ( F_1 + F_2 + F_3 )

- In general, resultant force ( F_R ) on a particle due to the presence of ( n ) number of surrounding particles is mathematically represented as –

F_R = ( F_1 + F_2 + F_3 + ..... + F_n )

*This is called principle of superposition of gravitational forces.*

__Newton’s Shell Theorem__

Newton’s shell theorem for gravitation gives gravitational force on a point mass due to a spherical shell or a solid sphere. Shell theorem for different cases is discussed as under –

### A. Gravitational Force outside shell

Consider that a point mass lies at point P outside of a uniform spherical shell as shown in figure (A).

- Shell has spherical symmetrical internal mass distribution.
- Then the shell attracts the point mass as if the entire mass of shell is concentrated at its centre of mass at O .

### B. Gravitational Force inside shell

Consider a point mass lies inside a uniform spherical shell as shown in figure (B).

- Let, ( r ) is the distance of point mass from the centre O of the shell.
- Now consider a imaginary spherical boundary of radius ( r ) .
- There is no mass interior to this spherical boundary.
- Therefore, gravitational force on the point mass will be zero.

### C. Gravitational Force outside sphere

Consider that a point mass lies at point P outside a solid sphere as shown in figure (C).

- Solid sphere has homogeneous and spherical symmetric internal mass distribution.
- Then the sphere attracts the point mass as if the entire mass of sphere is concentrated at its centre of mass at O .

### D. Gravitational Force inside sphere

Consider that a point mass lies inside a homogeneous solid sphere at a distance r from centre O of the sphere as shown in figure.

- The force on the point mass will act towards the centre of the sphere.
- This force is exerted by the spherical mass having radius ( r ) situated interior to the point mass.

__Gravitational Shielding__

*Gravitational shielding is the process of preventing a body or instruments from the action of gravitational force of attraction from nearby bodies.*

Consider about a particle of mass ( m_0 ) , which is kept inside a spherical shell of mass ( M ) . An other mass ( m_1 ) is present nearby outside the shell.

- By Newtons shell theorem, the gravitational force of attraction on the particle due to mass of the shell will be zero.
- But the particle will experience a force of attraction ( F ) due to the mass ( m_1 ) present outside the shell.
- Hence, the shell doesn’t shield the particle placed inside it from the gravitational force exerted by other mass outside of the shell.
- Thus gravitational shielding of a particle by placing inside a shell is not possible.

In this case the, gravitational force ( F ) acting on the particle is same as if shell is not present there.

Therefore, \quad F = G \left ( \frac { m_0 m_1 }{ r^2 } \right )

*Thus, gravitational shielding is not possible.*

See numerical problems based on this article.