Physics · Oscillations · NEET
Displacement is x = A sin(wt), which repeats once per period T (frequency f). But PE = (1/2) k x^2 = (1/2) k A^2 sin^2(wt). The square of a sine completes TWO cycles in the time the sine does one. Using the identity sin^2(wt) = (1 - cos(2wt))/2, the angular frequency inside becomes 2w. So energy oscillates at 2f while motion stays at f.
No. Total energy E = KE + PE = (1/2) k A^2 is CONSTANT in ideal SHM. It does not vary at f or 2f. Only KE and PE individually swing at 2f, but they always add up to the same fixed value. As KE goes down, PE goes up by the same amount.
If the body has time period T, then KE and PE each repeat every T/2. So their period is half the motion period, and their frequency is double. In one full oscillation of the body, KE reaches its maximum twice (at the two passes through the mean position) and PE reaches its maximum twice (at the two extreme positions).
Twice. The body passes through the mean position (x = 0) twice in one full cycle, once moving right and once moving left. At both passes speed is maximum, so KE is maximum both times. That is exactly why KE completes two cycles per motion cycle, giving frequency 2f.
No, this is a common mix-up. Velocity v = Aw cos(wt) varies at the SAME frequency f as displacement. It is the kinetic energy (which uses v^2) that varies at 2f. Squaring is what doubles the frequency, not the velocity itself.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
The frequency of PE is 2f. Its time period is T/2, half the period of the motion.
Because sin^2(wt) = (1 - cos(2wt))/2. The square of a sinusoid contains a term with angular frequency 2w, so the quantity oscillates twice as fast.
For ideal (undamped) SHM the rule KE and PE vary at 2f holds exactly. With light damping the amplitude slowly decays, but the fast oscillation of energy is still close to 2f.
PE is maximum at the two extreme positions (x = +A and x = -A), where the body is momentarily at rest. It reaches this maximum twice per cycle, which is why PE varies at 2f.
