Question

Figure 13-29 shows six paths by which a rocket orbiting a moon might move from point ato point b. Rank the paths according to (a) the corresponding change in the gravitational potential energy of the rocket–moon system and (b) the net work done on the rocket by the gravitational force from the moon, greatest first.

Short answer

1. The ranking of the paths according to the corresponding change in the gravitational potential energy of the rocket-moon system is $1=2=3=4=5=6$.
2. The ranking of the paths according to the corresponding net work done on the rocket by the gravitational force from the moon is $1=2=3=4=5=6$.

Step-by-step solution

The given data

There are six paths by which a rocket orbiting a moon move from point a to point b.

Understanding the concept of the potential energy and work

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. Since the change in gravitational potential energy and work done is independent of the path, we can rank the paths according to the initial and final position of an object.

Formulae:

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

The work done by the system due to energy change, $W=-∆U$ …(ii)

(a) Calculation of the rank of the paths according to the change in gravitational potential energy

The change in gravitational potential energy is independent of path.

From equations (i) and (ii) depends on the initial and final position of an object. For all paths the initial and final position is the same. Hence,the change in the gravitational potential energy of the rocket-moon systemis the samefor all paths.

Therefore, ranking of the paths according to the corresponding change in the gravitational potential energy of the rocket-moon system is $1=2=3=4=5=6$.

(b) Calculation of the rank of the paths according to the work done

The work done is independent of path considering equation (ii). It depends on the initial and final position which means the displacement of an object. For all paths, displacement of the rocket is the same. Hence, the net work done on the rocket by the gravitational force from the moonis the samefor all paths.

Therefore, ranking of the paths according to the corresponding net work done on the rocket by the gravitational force from the moon is $1=2=3=4=5=6$.