Figure 13-22 shows three arrangements of the same identical particles, with three of them placed on a circle of radius 0.20mand the fourth one placed at the center of the circle. (a) Rank the arrangements according to the magnitude of the net gravitational force on the central particle due to the other three particles, greatest first. (b) Rank them according to the gravitational potential energy of the four-particle system, least negative first.

The radius of the circle is, r = 0.20 m

The expression for the gravitational potential energy of the system of the particles is as follows,

$\mathbf{U}\mathbf{=}\frac{\mathbf{GMm}}{\mathbf{r}}$

Here, G is the gravitational constant, M is the mass of the heavy body, m is the mass of the light body, and R is the distance between two bodies.

The magnitude of the gravitational force in all three arrangements is the same because in all arrangements, the particles are identical and the distance of the particles on the circle from the particle at the center of the circle is the same. But, the directions of the forces are different.

Write expression for the gravitational force of attraction between two bodies.

$F=\frac{GMm}{R^2}$

Here, G is the gravitational constant, M is the mass of the heavy body, m is the mass of the light body, and R is the distance between two bodies.

In the first arrangement, the forces due to the upper and lower particles on the circle get canceled out and will be zero by considering the above equation.

So, the net force acting on the particle at the center is due to the third particle.

In the third arrangement, forces due to all three particles are parallel to each other. So, in this arrangement, the net force on the particle at the center will be the greatest as they add up by considering the above equation.

Thus, the rank according to the net gravitational force on the central particle is c > b > a.

It is known that all arrangements of the particles are identical and the distance of the particles on the circle from the particle at the center of the circle is the same. So, the gravitational potential energy in all those arrangements will be the same by considering the equation of gravitational potential energy.

Thus, the rank according to the gravitational potential energy on the central particle is a = b = c.