Q. 84.5( 12 Votes )

A comet orbits the 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 :

Suppose a comet orbits around sun in elliptical orbit , so its motion will be in accordance with Kepler's laws , now suppose at any instant planet was at a point A at a distance of r from sun and after a while it moved to a point B at a distance R’ from sun in this motion it subtended an angle 𝜽 with the sun and two position and its speed changed from v to v’ direction of motion is always tangential to orbit

The situation has been shown in figure

(a) Linear speed of the comet is not constant, and its speed increases when its closer to sun and decreases when its farther from sun and since, according to Kepler's second law comet must sweep equal areas in equal interval of time, i.e. area made between sun and line joining two positions of comet must be same in the same interval of time, but sun is not equidistant from planet so speed of planet changes so that area swept remains same.

(b) Now angular speed is also not constant as it is the rate of change of angle which is subtended between the two positions of comet and sun, since planet, speed changes so it covers unequal angles in equal interval of time so its angular speed varies.

(c) Angular momentum is conserved or same because there is no external torque acting on the comet as a force acting due to sun on a comet is parallel to line joining sun and planet or it is a central force so torque on a comet is zero and when no external torque acts on a body its angular momentum is conserved.

(d) Kinetic energy is not same or conserved as the speed of comet is varying and kinetic energy depends only on the speed of particle We know Kinetic Energy is given as

Where K is the kinetic energy of a body of mass moving with speed v, so as speed changes kinetic energy also changes

(e) Potential energy is also not conserved or same as it depends upon separation of two bodies and as the orbit is elliptical the distance between comet and sun changes continuously so potential energy is also not conserved as potential energy is given as

U = -Gm1m2/R

Where U is the potential Energy of a body of mass m1 at a point at a distance R from a center of mass of Body of mass m2, G is universal Gravitational constant

Since distance R of the comet of mass m1 changes from the sun of mass m2 changes continuously potential energy also changes.

(f) The total energy of the system is the sum of kinetic energy and potential energy and is conserved always according to the law of conservation of energy, so even kinetic energy and potential energy varies, the total energy remains same. i.e. both are interconverted to each other.

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