Consider the following statement for a particle moving in an elliptic orbit under the influence of a central force. Which of the following statement(s) is/are correct?a)The radius vector covers equal area in equal timeb)The motion takes place in a planec)The angular momentum is a constant of motiond)The path of a particle under the influence of a central force must lie in a plane.Correct answer is option 'A,B,C'. Can you explain this answer?

Question

Answer

Path of particle under the influence of central force can be anything... example : motion of an astroid coming to earth

Answer

We have angular momentum,

Answer

The correct answer is option 'A,B,C'.

Explanation:

a) The radius vector covers equal area in equal time:
In an elliptic orbit, the radius vector from the central force to the particle sweeps out equal areas in equal time intervals. This is known as Kepler's second law of planetary motion. It means that the particle moves faster when it is closer to the central force and slower when it is farther away. The area swept out by the radius vector in a given time interval is constant, indicating that the particle's speed is not constant.

b) The motion takes place in a plane:
The motion of a particle under the influence of a central force, such as gravity, takes place in a plane. This is known as the plane of motion. The force acting on the particle is always directed towards the center of the orbit, causing the particle to move in a two-dimensional plane. This can be observed in the motion of planets around the Sun or satellites around the Earth.

c) The angular momentum is a constant of motion:
Angular momentum is defined as the product of the moment of inertia and the angular velocity. In an elliptic orbit, the angular momentum of the particle remains constant. This is because the central force does not exert any torque on the particle, causing its angular momentum to be conserved. The conservation of angular momentum is a fundamental principle in physics and is applicable to various systems, including particles in elliptic orbits.

d) The path of a particle under the influence of a central force must lie in a plane:
This statement is incorrect. While the majority of particles under the influence of a central force move in a plane, there are exceptions. For example, comets have highly elliptical orbits that are not confined to a single plane. Additionally, particles in three-dimensional spaces can have non-planar orbits if they are subjected to multiple central forces or external torques. However, for the specific case of a single central force acting on a particle, its path will indeed lie in a plane.

In summary, the correct statements for a particle moving in an elliptic orbit under the influence of a central force are that the radius vector covers equal area in equal time, the motion takes place in a plane, and the angular momentum is a constant of motion.

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