Question

Derive an expression for energy in Simple Harmonic Motion (S.H.M).

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Answer 1

• Kinetic energy is also a periodic function of time, being zero when the displacement is maximum. Also, Kinetic energy is maximum when the particle is at the mean position. (T/2) is the period function of Kinetic energy.
• At the mean position, the potential energy is zero. At an extreme position, the potential energy is maximum.
• The total mechanical energy of a harmonic oscillation is independent of displacement or time and it depends on amplitude (i.e. maximum displacement).

The total energy of harmonic oscillation is conserved when there is no frictional force.

Answer 2

Expression for Energy in SHM :

KINETIC ENERGY:

$v = \dfrac{dx}{dt}$

$\implies \: v = \dfrac{d \{a \sin( \omega t) \}}{dt}$

$\implies \: v = a \omega \cos( \omega t)$

$\implies \: v = \omega \sqrt{ {a}^{2} \{1 - { \sin}^{2}( \omega t) \} }$

$\implies \: v = \omega \sqrt{ {a}^{2} - {x}^{2} }$

Now, KE will be :

$KE = \dfrac{1}{2} m {v}^{2}$

$\boxed{\implies \: KE = \dfrac{1}{2} m { \omega}^{2} ( {a}^{2} - {x}^{2} )}$

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Now, Potential Energy :

• Potential energy stored in the object will be equal to the work done to move the object with particular force.

$\displaystyle \: PE = \int \: dW$

$\implies \displaystyle \: PE = \int \:f \: dx$

$\implies \displaystyle \: PE = \int \:kx \: dx$

$\implies \displaystyle \: PE = \dfrac{1}{2} k {x}^{2}$

$\implies \displaystyle \: PE = \dfrac{1}{2} (m { \omega}^{2} ) {x}^{2}$

$\boxed{ \implies \displaystyle \: PE = \dfrac{1}{2} m { \omega}^{2} {x}^{2} }$

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Now, total energy will be sum of KE and PE:

$TE = KE + PE$

$\implies TE = \dfrac{1}{2}m { \omega}^{2}( {a}^{2} - {x}^{2} ) + \dfrac{1}{2} m { \omega}^{2} {x}^{2}$

$\boxed{ \implies TE = \dfrac{1}{2}m { \omega}^{2}{a}^{2} }$

Hope It Helps.