For a body executing S.H.M. :
(1) Potential energy is always equal to its K.E.
(2) Average potential and kinetic energy over any given time interval are always equal.
(3) Sum of the kinetic and potential energy at any point of time is constant.
(4) 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 :

Solution

Physics sir · 3 steps

KE goes as cos² ω t and PE as sin² ω t: fixed sum, equal averages only over a full period.

1. Write both energies. Take x = A sin ω t, so v = Aω cos ω t. With k = mω², both energies carry the same factor 1/2kA².

   K = 1/2kA²cos² ω t, U = 1/2kA²sin² ω t, K + U = 1/2kA²

2. Averages over a period. Over one full period, the average of sin² ω t and the average of cos² ω t are both 1/2. So ⟨ K ⟩ = ⟨ U ⟩ = 1/4kA².

   Statement	Check	True?
   (1) U = K always	Equal only where sin² ω t = cos² ω t, that is at x = ± A/√2	No
   (2) Averages equal over any interval	A short interval near an extreme has K ≈ 0 while U is near its maximum	No
   (3) K + U constant	1/2kA²(cos² ω t + sin² ω t) = 1/2kA²	Yes
   (4) Averages equal over one period	Both averages are 1/4kA²	Yes

3. So. Statements (3) and (4) are correct, which is option (a).

The idea to keep. In SHM total energy stays at 1/2kA² and keeps moving between KE and PE. They share it equally only on average over whole periods, never moment by moment.

Answer from the NTA final answer key. Solution by PrepWiser.