Question

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.

Hint:

A central potential implies rotational symmetry about the center, ensuring angular momentum conservation and planar motion.

Answer 1

The Correct Option is A

Step 1: Understanding central potential.
A central potential depends only on the distance \(r\) from a fixed point (the center), i.e., \(V = V(r)\). The force is always directed toward or away from the center.

Step 2: Conservation of angular momentum.
The torque on the particle is zero because \(\vec{r} \times \vec{F} = 0\). Hence, angular momentum \(\vec{L} = \vec{r} \times \vec{p}\) is conserved in both magnitude and direction.

Step 3: Implication for motion.
Because \(\vec{L}\) is constant in direction, the position vector \(\vec{r}\) always lies in a fixed plane perpendicular to \(\vec{L}\). Therefore, the motion is confined to a plane.

Step 4: Conclusion.
The motion is restricted to a plane due to conservation of angular momentum.