Particle moving under central force

When a particle is moving under a central force, it means that the force acting on the particle is directed towards or away from a fixed point in space, known as the center. This force can be either attractive or repulsive, depending on the nature of the central force.

Mass, energy, and angular momentum

In this scenario, the mass of the particle is denoted by m, the energy is denoted by E, and the angular momentum is denoted by L. These quantities play crucial roles in determining the behavior of the particle.

Turning point

A turning point is a point in the path of the particle where its motion changes from inward to outward or vice versa. At the turning point, the radial velocity of the particle becomes zero.

Speed of the particle at the turning point

To determine the speed of the particle at the turning point, we can use the conservation of energy and angular momentum.

Conservation of energy

The conservation of energy states that the total energy of the particle remains constant throughout its motion. This can be expressed as:

E = (1/2)mv^2 + V(r)

Where v is the velocity of the particle, and V(r) is the potential energy function associated with the central force.

Conservation of angular momentum

The conservation of angular momentum states that the angular momentum of the particle remains constant throughout its motion. This can be expressed as:

L = mvr

Where r is the distance of the particle from the center.

Deriving the speed at the turning point

At the turning point, the radial velocity of the particle becomes zero. Therefore, we can set v = 0 in the equation for the conservation of angular momentum:

L = mvr = 0

This implies that either m = 0 (which is not physically possible) or r = 0. Since r cannot be zero, we conclude that m = 0.

Conclusion

The speed of the particle at the turning point is zero. This means that the particle comes to a momentary stop at the turning point before changing its direction of motion.