Question: (a) In Problem 77, remove sphere Aand calculate the gravitational potential energy of the remaining three-particle system. (b) If Ais then put back in place, is the potential energy of the four-particle system more or less than that of the system in (a)? (c) In (a), is the work done by you to remove Apositive or negative? (d) In (b), is the work done by you to replace Apositive or negative?

The mass of the sphere A with coordinates is$m_A=40kg,\left(0,50cm\right)$

The mass of the sphere B with coordinates is$m_B=35kg,\left(0,0cm\right)$

The mass of the sphere C with coordinates is$m_c=200kg,\left(-80cm,0\right)$

The mass of the sphere D with coordinates is$m_D=50kg,\left(40cm,0\right)$

The energy that an item has or acquires when its location changes as a result of being in a gravitational field is known as gravitational potential energy.

We can use the concept of gravitational potential energy to find the energy of the given system.

Formula:

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

Where ,

G is the potential energy

U is the gravitational constant ()

M is the mass of earth

m is the mass of body

r is the distance of body from the center of earth

Gravitational potential energy of the three particle system:

The distance between the sphere C and D sphere is$r_{CD}=1.2m$

The total gravitational potential energy of the system is

$$\begin{gathered} U=-\frac{G{M}_B{M}_C}{r_C}-\frac{G{M}_B{M}_D}{r_D}-\frac{G{M}_C{M}_D}{r_{CD}} \\ U=-\frac{\left(6.67\times {10}^{-11}\frac{{\text{Nm}}^{\text{2}}}{{\text{kg}}^{\text{2}}}\right)\times 35\text{kg}\times 200\text{kg}}{0.80\text{m}}-\frac{\left(6.67\times {10}^{-11}\frac{{\text{Nm}}^{\text{2}}}{{\text{kg}}^{\text{2}}}\right)\times 35\text{kg}\times 50\text{kg}}{0.40\text{m}} \\ -\frac{\left(6.67\times {10}^{-11}\frac{{\text{Nm}}^{\text{2}}}{{\text{kg}}^{\text{2}}}\right)\times 200\text{kg}\times 50\text{kg}}{1.2\text{m}} \end{gathered}$$

$U=-1.43\times {10}^{-6}J$

If sphere A is put back, compare the potential energy of the four particle system:

The gravitational potential energy is a negative term. When we put the sphere, it lowers the total potential energy of the system.

When we remove the sphere A it increases the total energy of the system; hence, it is positive work done.

When we replace the sphere A, it decreases the total energy of the system; hence, it is negative work done. The system becomes more negative.