Question

For a body executing S.H.M. :
(a) Potential energy is always equal to its K.E.
(b) Average potential and kinetic energy over any given time interval are always equal.
(c) Sum of the kinetic and potential energy at any point of time is constant.
(d) Average K.E. in one time period is equal to average potential energy in one time period.
Choose the most appropriate option from the options given below :

• A. only (b)
• B. (b) and (c)
• C. only (c)
• D. (c) and (d)

Hint:

In SHM, Energy oscillates with twice the frequency of the displacement. Remember that the average P.E. and average K.E. over one full cycle are both exactly half of the total energy.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Simple Harmonic Motion (SHM) involves a continuous transformation between kinetic energy (K.E.) and potential energy (P.E.), while their sum remains constant in the absence of dissipative forces.
Step 2: Detailed Explanation:
(a) Potential energy \(U = \frac{1}{2} k x^2\) and Kinetic energy \(K = \frac{1}{2} k (A^2 - x^2)\). They are equal only when \(x = \pm \frac{A}{\sqrt{2}}\), not always. So (a) is false.
(b) Average values are only guaranteed to be equal over a full period or specific symmetric intervals. "Any given interval" is too broad. So (b) is false.
(c) Total Energy \(E = U + K = \frac{1}{2} k A^2\). This is constant throughout the motion. So (c) is true.
(d) Over a full time period \(T\), both the average potential energy and average kinetic energy are equal to \(\frac{1}{4} k A^2\). So (d) is true.
Step 3: Final Answer:
Statements (c) and (d) are correct.