Published by: sadikshya
Published date: 21 Jun 2021
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Fig. The energy in Simple Harmonic Motion
The total energy of an oscillating particle in simple harmonic motion is the sum of its KE and PE if a 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
The kinetic energy of the particle of mass m is,
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From the 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,
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This work done is stored in the body as potential energy.
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Thus we find that the total energy of a particle executing simple harmonic motion is ½ mω2a2.
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Fig. Energy Displacement Diagram
(i) When the particle is at the mean position y = 0, from Eqn (1) it is known that kinetic energy is maximum and from Eqn. (2) it is known that potential energy is zero. Hence the total energy is wholly kinetic.
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(ii) When the particle is at the extreme position y = +a, from equation (1), it is known that kinetic energy is zero and from Eqn. (2) it is known that Potential energy is maximum. Hence the total energy is wholly potential.
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(iii) When y = a/2,
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If the displacement is half of the amplitude, K = ¾ E and U = ¼ E. K and U are in the ratio 3: 1.
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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.