5.1 Gravitational potential energy for a system of particles

1.1 Characteristics of gravitational force
1.2 Universal gravitational constant
1.3 Principle of superposition of gravitation
1.4 Gravity
1.5 Acceleration due to gravity
1.6 Relation between $g$ and $G$

2.1 Variation of acceleration due to gravity $(g)$ due to shape of the earth
2.2 Variation of acceleration due to gravity $(g)$ due to altitude
2.3 Variation of acceleration due to gravity $(g)$ due to depth
2.4 Variation of acceleration due to gravity $(g)$ due to rotation of earth

3.1 Gravitational field due to a point mass
3.2 Gravitational field due to a uniform solid sphere
3.3 Gravitational field due to a uniform spherical shell
3.4 Gravitatioal field due to a uniform circular ring at a point on its axis

4.1 Gravitational potential due to a point mass
4.2 Gravitational potential due to a uniform solid sphere
4.3 Gravitational potential due to a uniform thin spherical shell
4.4 Gravitational potential due to a uniform ring at a point on its centre
4.5 Relation between gravitational field and gravitational potential

5.1 Gravitational potential energy for a system of particles
5.2 Gravitational potential energy of a body on earth's surface

6.1 Orbital speed of satellite
6.2 Time period of a satellite
6.3 Angular momentum of a satellite
6.4 Energy of a satellite
6.5 Types of satellite
6.6 Binding energy
6.7 Escape velocity
6.8 Weightlessness

7.1 Kepler's first law
7.2 Kepler's second law
7.3 Kepler's third law

The gravitational potential energy of a system of particles is equal to the sum of gravitational potential energy for each pair of particles.

Let us consider a system of $4$ particles having masses $m_1$, $m_2$, $m_3$ and $m_4$ respectively.

Number of pair can be formed is $N$, $$N = {}^n{C_2}$$$$N = {}^4{C_2}$$$$N = 6$$

where,

$n$: Number of particles

$N$: Number of pairs

So, the gravitational potential energy for a system of particles is given by,

$$U = - G\left[ {\frac{{{m_1}{m_2}}}{{{r_{12}}}} + \frac{{{m_2}{m_3}}}{{{r_{23}}}} + \frac{{{m_3}{m_4}}}{{{r_{34}}}} + \frac{{{m_4}{m_1}}}{{{r_{41}}}} + \frac{{{m_1}{m_3}}}{{{r_{13}}}} + \frac{{{m_2}{m_4}}}{{{r_{24}}}}} \right]$$

where ${{r_{12}}}$, ${{r_{23}}}$, ${{r_{34}}}$, ${{r_{41}}}$, ${{r_{13}}}$ and ${{r_{24}}}$ is the distance between the respective particles.