4.3 Gravitational potential energy  (Page 3/4)

Here,

$$U_1=0$$

$$U_2=U\left(\mathrm{say}\right)$$

$$r_1=\infty$$

$$r_2=r\left(\mathrm{say}\right)$$

Putting values in the expression of the change of potential energy, we have :

$$\Rightarrow U-0=G{m}_1{m}_2[\frac{1}{\infty }-\frac{1}{r}]$$

$$\Rightarrow U=-\frac{G{m}_1{m}_2}{r}$$

By definition, this potential energy is equal to the negative of work by gravitational force and equal to the work by an external force, which does not produce kinetic energy while the particle of mass “ $m_2$ ” is brought from infinity to a position at a distance “r” from other particle of mass “ $m_1$ ”.

We see here that gravitational potential energy is a negative quantity. As the particles are farther apart, "1/r" becomes a smaller fraction. Potential energy, being a negative quantity, increases. But, the magnitude of potential energy becomes smaller. The maximum value of potential energy is zero for r = ∞ i.e. when particles are at very large distance from each other

On the other hand, the fraction "1/r" is a bigger fraction when the particles are closer. Gravitational potential energy, being a negative quantity, decreases. The magnitude of potential energy is larger. This is consistent with the fact that particles are attracted by greater force when they are closer. Hence, if a particles are closer, then it is more likely to be moved by the gravitational force. A particle away from the first particle has greater potential energy, but smaller magnitude. It is attracted by smaller gravitational force and is unlikely to be moved by gravitational force as other forces on the particle may prevail.

Gravitational potential energy of a system of particles

We have formulated expression for the gravitational potential energy of two particles system. In this section, we shall find gravitational potential energy of a system of particles, starting from the beginning. We know that zero gravitational potential energy is referred to infinity. There will no force to work with at an infinite distance. Since no force exists, no work is required for a particle to bring the first particle from infinity to a point in a gravitation free region. So the work by external force in bringing first particle in the region of zero gravitation is "zero".

What about bringing the second particle (2) in the vicinity of the first particle (1)? The second particle is brought in the presence of first particle, which has certain mass. It will exert gravitational attraction on the second particle. The potential energy of two particles system will be given by the negative of work by gravitational force due to particle (1)as the second particle is brought from infinity :

$$U_{12}=-\frac{G{m}_1{m}_2}{r_{12}}$$

We have subscripted linear distance between first and second particle as “ $r_{12}$ ”. Also note that work by gravitational force is independent of the path i.e. how force and displacement are oriented along the way second particle is brought near first particle.

Now, what about bringing the third particle of mass, “ $m_3$ ”, in the vicinity of the first two particles? The third particle is brought in the presence of first two particles, which have certain mass. They will exert gravitational forces on the third particle. The potential energy due to first particle is equal to the negative of work by gravitational force due to it :