Question

The potential energy of a simple harmonic oscillator of at position half way from mean position to extreme position is what fraction of its total energy

Answer 1

Answer:

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Energy in Simple Harmonic Motion

Simple Harmonic Motion

The total energy (E) of an oscillating particle is equal to the sum of its kinetic energy and potential energy if conservative force acts on it.

The velocity of a particle executing SHM at a position where its displacement is y from its mean position is v = ω√a2 – y2

Kinetic energy

Kinetic energy of the particle of mass m is,

K = ½ m [ω√a2 – y2]2

…... (1)

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 work done for the displacement y is,

W = \int dW = \int_{0}^{y}ky dy

As k = mω2, therefore,

W =\int_{0}^{y}m\omega ^{2}y dy

Thus, W = ½ mω2y2 …... (2)

This work done is stored in the body as potential energy.