Question

At perihelion, the gravitational potential energy of Pluto in its orbit has
A) a maximum value.
B) a minimum value.
C) the same value as at every other point in its orbit.
D) a value that depends on the sense of rotation.

Answer 1

Hint: At perihelion, the Pluto will be closest to the Sun. The gravitational potential energy between two masses depends on the distance between the two masses. As the distance of separation decreases, the potential energy increases but it is a negative value.

Formula used:
The gravitational potential energy between two masses is given by, $V = \dfrac{{ - GMm}}{r}$ where $G$ is the gravitational constant, $M$ and $m$ are the masses and $r$ is the distance between the two masses.

Complete step by step answer:
Step 1: Sketching a figure describing the revolution of Pluto around the Sun.
Pluto’s orbit of revolution has an elliptic nature. The Sun is considered to be located at one of the foci of the elliptical orbit.
The figure given below describes the revolution of Pluto around the Sun.

Step 2: Expressing the relation for the gravitational potential energy of Pluto.
The gravitational potential energy of Pluto is given by, $V = \dfrac{{ - GMm}}{r}$ where $G$ is the gravitational constant, $M$ is the mass of the Sun, $m$ is the mass of Pluto and $r$ is the distance between the Sun and Pluto.
As Pluto revolves around the Sun, the only quantity in the above equation that changes is the distance of separation $r$.
So we have $V \propto \dfrac{{ - 1}}{r}$ .

Since the distance of separation will be minimum at perihelion, the value of gravitational potential energy is most negative or minimum when Pluto is closest to the Sun. So the correct option is B.

Note:
From the relation for the gravitational potential energy, it is clear that it has the highest value when the distance of separation $r$ is the least. But the presence of the negative sign indicates that the highest value will be in fact the highest negative value. Aphelion refers to the point in the orbit when the distance of separation is the least.