Question

A comet orbits a sun in a highly elliptical orbit. Does the comet have a constant (a) linear speed, (b) angular speed, (c) angular momentum, (d) kinetic energy, (e) potential energy, (f) total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the sun.

Answer 1

Hint:The comet revolves around the sun in elliptical orbit. The force acting on the comet which keeps the comet in orbit is the gravitational force of attraction. The gravitational force of attraction is an internal force. So, on the comet and sun system there is no external force acting.

Complete step-by-step solution:
When external torque acting on the system is zero then the angular momentum of the system remains constant.
Using Newton’s law of rotational motion,
$\overrightarrow{\tau }=\dfrac{d}{dt}\left[ \overrightarrow{L} \right]$
Where,
$\overrightarrow{\tau }=$net external torque acting on the system.
$\dfrac{d}{dt}\left[ \overrightarrow{L} \right]=$the rate of change of angular momentum
If the net external torque acting on the system is zero then the rate of the change of angular momentum is zero. Hence, the angular momentum of the system is constant.
A comet while going on an elliptical orbit around the Sun has constant angular momentum due to no external torque and total energy due to Law of conservation of energy at all locations but other quantities vary with locations.

(a)
Angular momentum, $L=mvr=\text{constant}$
Where,
$r$ is the distance of the comet from the sun which is not constant in the elliptical path.
As angular momentum and the mass of the comet is constant, so to compensate the change in value of $r$ the linear speed of the comet changes. Hence, the linear speed of the comet is not constant.

(b)
Angular momentum, $L=m\omega {{r}^{2}}=\text{constant}$
Where,
$\omega $ is the angular speed of the comet.
As angular momentum and the mass of the comet is constant, so to compensate for the change in value of $r$ the angular speed of the comet changes. Hence, the angular speed of the comet is not constant.

(c)
Angular momentum is constant because there is no external torque.

(d)
Since speed is not constant, Kinetic Energy is not constant.

(e)
Total energy is constant by law of conservation of energy.
\[T.E.=P.E.+K.E.=constant\]
Since \[K.E.\] is not constant, hence, \[P.E.\] is not constant.

(f) Total energy is constant by law of conservation of energy.

Therefore,
Option (f) is the correct answer.


Note:->Total energy of the system remains constant unless the external agent does work on the system.
->Total angular momentum of the system remains constant unless the external agent applies torque on the system.