Question

For a particle moving in a general central force field, which of the following statement(s) is/are true?

• A. The angular momentum is a constant of motion
• B. Kepler’s second law is valid
• C. The motion is confined to a plane
• D. Kepler’s third law is valid

Answer 1

The Correct Option is A, B, C

In this problem, we need to determine which statements are true for a particle moving in a general central force field. Let's analyze each statement:

1. The angular momentum is a constant of motion:

   In a central force field, the force acting on the particle is always directed towards or away from a fixed point (the center). Accordingly, there is no torque about this point, meaning the angular momentum of the particle is conserved. Therefore, this statement is true.

2. Kepler’s second law is valid:

   Kepler's second law, also known as the law of areas, states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. In the context of a central force field, this law is valid due to the conservation of angular momentum, leading to an areal velocity that is constant. Hence, this statement is true.

3. The motion is confined to a plane:

   For a particle subjected to a central force, the motion remains planar. This is because the cross product of position and velocity vectors, which determines the direction of the angular momentum, remains constant. Hence, the particle continues to move in a single plane. Thus, this statement is true.

4. Kepler’s third law is valid:

   Kepler's third law relates the period of orbit to the semi-major axis of the orbit, formulated specifically for gravitational forces between celestial bodies. For a general central force field, this law might not universally apply unless the force follows the inverse square law, like gravity. Hence, this statement might not be generally valid.

Based on the above reasoning, the correct statements are:

• The angular momentum is a constant of motion
• Kepler’s second law is valid
• The motion is confined to a plane

Thus, the correct answers align with the provided information. Kepler’s third law is not universally valid in all central force fields, thereby being the exception in this case.