Question

In a simple harmonic motion
(a) the potential energy is always equal to the kinetic energy
(b) the potential energy is never equal to the kinetic energy
(c) the average potential energy in any time interval is equal to the average kinetic energy in that time interval
(d) the average potential energy in one time period is equal to the average kinetic energy in this period.

Answer 1

Answer ⇒ Option (d). The average potential energy in one time period is equal to the average kinetic energy in this period.

Explanation ⇒ The sum of Kinetic Energy and Potential Energy is constant, I mean Total Energy of the system is constant but it is not necessary that they are equal. Therefore, Option (a) is not true.

Now,  their sum, i.e. the total energy of the Simple Harmonic Motion,  is constant at every instant instant, so they may be equal at a single instant.

Kinetic Energy is maximum at the mean position while Potential Energy is minimum at the extreme position.  Now, Average Potential and kinetic Energy cannot be equal in a given time interval.  It will be equal if the no. of osculations made will be in  whole number ratio.

The Change of the Kinetic and Potential energy is exactly same in one time period except that K.E. is maximum at the mean position while P.E. is maximum at the extremes.

So the average P.E. and K.E. are equal in one time period.

Hence, Option (d) is true.

Hope it helps.