Question

Rank the four systems of equal mass particles shown in check point 2 according to the absolute value of the gravitational potential energy of the system, greatest first.

Short answer

The ranking of the systems according to the absolute value of the gravitational potential energy is 1 > 2 = 4 > 3.

Step-by-step solution

The given data 

The figure for four systems of equal mass particles is given.

Understanding the concept of the gravitational potential energy

When an object is present in a gravitational field, it has or can gain gravitational potential energy, which is energy that results from a change in position. We can find the potential energy of each system of the particles. After comparing them, we can rank them according to the absolute value of the gravitational potential energy.

Formula:

The gravitational potential energy between two bodies of masses M and m separated by distance r is, $U=\frac{-GMm}{r}$ …(i)

Calculation of the rank according to the potential energy

Let the mass of each particle be m.

The gravitational P.E of system 1 is given using equation (i) as follows:

$P.E=\frac{G{m}^2}{d}-\frac{G{m}^2}{D}-\frac{G{m}^2}{(D-d)}$

The gravitational P.E of system 2 is given using equation (i) as follows:

$P.E=\frac{G{m}^2}{d}-\frac{G{m}^2}{D}-\frac{G{m}^2}{\sqrt{d^2-D^2}}$

The gravitational P.E of system 3 is given using equation (i) as follows:

$P.E=\frac{G{m}^2}{d}-\frac{G{m}^2}{D}-\frac{G{m}^2}{d+D}$

The gravitational P.E of system 4 is given using equation (i) as follows:

$P.E=\frac{G{m}^2}{d}-\frac{G{m}^2}{D}-\frac{G{m}^2}{\sqrt{d^2+D^2}}$

All the above equations have the same first two terms. It is the last term in the equations which makes the difference. We can see that the 1^st equation has the smallest denominator and the 3rd equation has the greatest denominator. The value of the denominator in the 2nd and 4^th equations is the same; it is greater than the 1^st equation and smaller than the 3^rd equation.

Hence, the rank of the positions according to their gravitational potential energies is 1 > 2 = 4 > 3.