Question

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?

Short answer

Answer:

1. The gravitational potential energy of the remaining three particle system is$U=-1.43\times {10}^{-6}J$
2. If sphereis put back, it lowers the potential energy of the four particle system
3. Work done by you to remove the sphereis positive
4. Work done by you to replace the sphere is negative

Step-by-step solution

Identification of given data

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$m_D=50kg,\left(40cm,0\right)$

Significance of gravitational potential energy

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

(a) Determining the gravitational potential energy of the remaining three particle system

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

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

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

(b) If sphere is put back, the potential energy of the four particle system

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.

(c) Work done by you to remove the sphere  

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

(d) Work done by you to replace the sphere 

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.