Question

Derive the expression for the kinetic and potential energy of harmonic oscillator.Hence show that total energy is conserved in SHM. Draw graphs for (1) energy vs time and (2)energy vs displacement

Answer 1

For SHM,
acceleration , a = -ω²y
F = ma = -mω²y
now, work , W = F.dy cos180° { because displacement and acceleration are in opposite direction so, cos180° taken }
W = ∫mω²y.dy = mω²y²/2

use standard form of SHM , y = Asin(ωt ± Ф)
W = mω²A²/2 sin²(ωt ± Ф)
We know, Potential energy is work done stored in system .
so, P.E = W = mω²A²/2 sin²(ωt ± Ф)

again, Kinetic energy , K.E = 1/2mv² , here v is velocity
we know, v = ωAcos(ωt ± Ф)
so, K.E = mω²A²/2cos²(ωt ± Ф)

Total energy = K.E + P.E
= mω²A²/2 cos²(ωt ± Ф) + mω²A²/2 sin²(ωt ± Ф)
= mω²A²/2 [ cos²(ωt ± Ф) + sin²(ωt ± Ф) ] = mω²A²/2 [ ∵sin²α + cos²α = 1]
= mω²A²/2 = constant

Hence, total energy is always constant .

Answer 2

$\textbf{Answer is in Attachment !!}$