For a particle moving in a central potential, which one of the following statements is correct?a)The motion is restricted to a plane due to the conservation of angular momentumb)The motion is restricted to a plane due to the conservation of energy onlyc)The motion is restricted to a plane due to the conservation of linear momentumd)The motion is not restricted to a planeCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Explanation:

The motion of a particle in a central potential refers to the motion of a particle under the influence of a force that depends only on its distance from a fixed point (the center of the potential).

Conservation of Angular Momentum:
Angular momentum is defined as the cross product of the particle's position vector and its linear momentum. For a central potential, the force acting on the particle is always directed radially, towards or away from the center. Since the force is always perpendicular to the position vector, it does not exert any torque on the particle. Thus, angular momentum is conserved.

Conservation of Energy:
In a central potential, the force acting on the particle is conservative, meaning it can be derived from a potential energy function. This implies that the total mechanical energy of the particle, given by the sum of its kinetic and potential energies, is conserved. The conservation of energy does not impose any restrictions on the motion of the particle.

Conservation of Linear Momentum:
Linear momentum is defined as the product of the particle's mass and velocity. In a central potential, the force acting on the particle is always directed radially and depends only on the distance from the center. Therefore, the force does not change the direction of the particle's velocity, resulting in the conservation of linear momentum.

Motion Restricted to a Plane:
Since the angular momentum of the particle is conserved in a central potential, and angular momentum is defined as the cross product of the position vector and linear momentum, it implies that the position vector and linear momentum must lie in the same plane. This means that the motion of the particle is restricted to a plane. Therefore, option 'A' is correct.

Conclusion:
The motion of a particle in a central potential is restricted to a plane due to the conservation of angular momentum. The conservation of energy and linear momentum does not impose any restrictions on the motion of the particle.

Answer

For central force problem angular momentum J is conserved and r.J =0 which ensure that motion of particle is confined in plane

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