Energy in SHM , Kinetic Energy , Potential Energy

Kinetic Energy (K.E)

From definition of kinetic energy,
$ \displaystyle K = \frac{1}{2}mv^2 $

Let , x = Asinωt

On differentiating with respect to time ,

$ \displaystyle \frac{dx}{dt} = \omega A cos\omega t $

Hence , v = ω A cosωt

$ \displaystyle K = \frac{1}{2}m\omega^2 A^2 cos^2\omega t $

$ \displaystyle K = \frac{1}{4}m\omega^2 A^2 (1 + cos2\omega t) $

Kinetic energy varies periodically with double the frequency of SHM

$ \displaystyle v = \omega (\sqrt{A^2 – x^2} )$

$ \displaystyle K = \frac{1}{2}m\omega^2 (A^2 – x^2 ) $

for x = 0,

$ \displaystyle K = \frac{1}{2}m\omega^2 A^2 $ = Kmax

for x = A, K = 0 = Kmin

Potential Energy (PE)

Potential Energy U is given by

$ \displaystyle  \int dU = -\int dW = -\int \vec{F}.\vec{dx}$

$ \displaystyle U = -\int_{0}^{x}m\omega^2 x dx cos180 $

$ \displaystyle U = \frac{1}{2}m\omega^2 x^2 $

In above derivation

(i) Angle between F and dx is taken as 180° as the two are oppositely directed.

(ii) Reference zero for U is taken at x = 0.

$ \displaystyle U = \frac{1}{2}m\omega^2 A^2 sin^2\omega t $

$\displaystyle U = \frac{1}{4}m\omega^2 A^2 (1 – cos2\omega t) $

Like Kinetic energy, Potential energy also varies periodically with double the frequency of SHM

$ \displaystyle U = \frac{1}{2}m\omega^2 x^2 $

at x = 0, U = 0 = Umin

for x = A ,

$ \displaystyle U = \frac{1}{2}m\omega^2 A^2 $

Total Mechanical Energy in SHM

Total Mechanical Energy , E = K + U = $ \displaystyle U = \frac{1}{2}m\omega^2 A^2 $ = constant

The variation of K , U and E as a function of displacement and time are shown below.

If y = A sin ωt ;

then $ \displaystyle K = \frac{1}{4}m\omega^2 A^2 (1 + cos2\omega t) $

and $ \displaystyle U = \frac{1}{4}m\omega^2 A^2 (1 – cos2\omega t) $

=> E = K + U = $ \displaystyle \frac{1}{2}m\omega^2 A^2 $ = constant

i.e. E does not depend upon time or x.

The variation of K, U and E as function of position and time are shown below :


Also Read :

→ Stable , Unstable & Neutral Equilibrium
→ S.H.M :Linear SHM & Angular SHM
→ Analytical Treatment to SHM
→ Average Value of P.E. & K.E. of Harmonic Oscillator
→ SHM as a projection of circular motion
→ Simple Pendulum in Inertial & Non Inertial Frame
→ Time period of a Long Pendulum
→ SHM of Spring Mass System
→ Physical Pendulum & Torsional Pendulum
→ Undamped & Damped simple harmonic oscillations

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