Question

Drive an expression for Kinetic and potentical
energy of an harmonic oscillation have
Show the total energy is conserved in Shin
Draw a graph for energy​

Answer 1

SOLUTION

$we \: know \: that \: for \: harmonic \: motion \\ \\ x = A \sin(wt) \\ \\ du= - f.dx \: \\ where \: du \: is \: the \: potential \\ energy \: of \: the \: particle \\ \\ for \:simple \: harmonic \: motion \: f = - kx \\ \\ du = kx.dx \\ after \: indefinite \: integration \: we \: get \\ \\ u = \frac{1}{2} k {x}^{2} \\ \\ u = \frac{1}{2} k {(A \sin(wt) })^{2} .........(1) \\ \\ v = Aw \cos(wt) \\ \\ kinetic \: energy \: = \frac{1}{2} m {v}^{2} \\ kinetic \: energy = \frac{1}{2} m {(Aw \cos(wt) )}^{2} ......(2) \\ \\ add \: (1) \: and \: (2) \\ total \: energy = \frac{1}{2} k {A}^{2} {sin}^{2}( wt) + \frac{1}{2} {m} {w}^{2} {A}^{2} {cos}^{2} (wt) \\ \\ for \: simple \: harmonic \: motion \\ \\ k = m {w}^{2} \\ \\ we \: get \: \\ \\ total \: energy = \frac{1}{2} m {w}^{2} {A}^{2} ( {cos}^{2} (wt) + {sin}^{2} (wt)) \\ \\ total \: energy \: = \frac{1}{2} m {w}^{2} {A}^{2} .....which \: is \: constant \\ or \: always \: remain \: same$