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 momentum.b)The motion is restricted to a plane due to the conservation of energy only.c)The motion is restricted to a plane due to the conservation of linear momentum.d)The motion is not restricted to a plane.Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Explanation:

Central Potential:
A central potential is a type of potential energy that depends only on the distance between the particle and a fixed point in space. The potential energy is spherically symmetric, meaning it is the same at every point that is equidistant from the fixed point.

Conservation of Angular Momentum:
Angular momentum is a vector quantity that measures the rotational motion of a particle or system of particles. In the case of a central potential, the angular momentum of the particle is conserved because there is no external torque acting on it.

Conservation of Energy:
Energy is a scalar quantity that measures the ability to do work or cause change. In the case of a central potential, the energy of the particle is conserved because there is no external force doing work on it.

Motion Restricted to a Plane:
When the angular momentum is conserved, it means that the particle's motion is confined to a plane. This is because the direction of the angular momentum vector is perpendicular to the plane of motion. As a result, the particle moves in a plane defined by the initial conditions.

Explanation of Answer:
The correct answer is option 'A' - The motion is restricted to a plane due to the conservation of angular momentum.

When a particle moves in a central potential, the conservation of angular momentum restricts its motion to a plane. This can be understood by considering the following:

- The particle's angular momentum vector is conserved because there is no external torque acting on it. This means that the magnitude of the angular momentum remains constant, as well as its direction.

- The direction of the angular momentum vector is perpendicular to the plane of motion. This is because the angular momentum is defined as the cross product of the position vector and the linear momentum vector. The cross product of two vectors is perpendicular to both vectors.

- Since the direction of the angular momentum vector is fixed, the particle's motion is confined to a plane defined by the initial conditions. The particle can move freely within this plane, but it cannot move out of it.

Therefore, the correct statement is that the motion of a particle in a central potential is restricted to a plane due to the conservation of angular momentum.

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