Question

Derive an expression for kinetic and potential energies of a simple harmonic oscillators. Hence show that total energy is conserved in simple harmonic motion.In which position of the oscillators is the energy wholy kinetic or potential.

Answer 1

If we talk About KINETIC ENERGY ――

Kinetic energy of the particle of mass m is,

K = ½ m [ω√a2 – y2]2

Kinetic energy of the particle of mass m is,

K = ½ m [ω√a2 – y2]2


Potential energy

From definition of SHM F = –ky the work done by the force during the small displacement dy is dW = −F.dy = −(−ky) dy = ky dy


Total energy, E = K+U

= ½ mω2 (a2 – y2) + ½ mω2y2

= ½ mω2a2

So,
Thus we find that the total energy of a particle executing simple harmonic motion is ½ mω2a2.
 the displacement is half of the amplitude, K = ¾ E and U = ¼ E. K and U are in the ratio 3 : 1.

E = K+U = ½ mω2a2

At any other position the energy is partly kinetic and partly potential.

This shows that the particle executing SHM obeys the law of conservation of energy.

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