Question

A ball is attached to a vertical spring. The ball is initially supported at a height y so that the spring is neither stretched nor compressed. The ball is then released from rest and it falls to a height y - h before moving upward. Consider the following quantities: translational kinetic energy, gravitational potential energy, elastic potential energy. When the ball was at a height y - (h/2), which of the listed quantities has values other than zero joules? Please explain.

Answer 1

The ball performs SHM with height y and (y-h) as extreme points. Equilibrium point lies midway between the extreme points. So, equilibrium position is at height y-h/2.

In SHM, at the equilibrium position, velocity is maximum and hence, not 0. Kinetic energy=1/2mv² where m is mass of the object and v is its velocity. Here,velocity is non zero, hence kinetic energy is non zero.

Gravitational potential energy is given by mgH, where m is mass of the object, g is gravitational acceleration, H is height of the object. Here, H=y-h/2 which is non-zero. Hence, potential energy is also non-zero.

Elastic potential energy of a spring=1/2kx² where k is spring constant and x is displacement from natural length.In the given problem,spring is at natural length at height y. So, at height y-h/2 , spring is elongated by a magnitude h/2 which is non-zero. Hence, elastic potential energy is also non-zero.

So, all of the listed quantities have non-zero values at height y-h/2