Average Value of Potential Energy , Kinetic Energy of Harmonic Oscillator

The average value of P.E. for complete cycle is given by

$ \displaystyle U_{avg}= \frac{1}{T}\int_{0}^{T}U dt $

$ \displaystyle U_{avg}= \frac{1}{T}\int_{0}^{T}\frac{1}{2}m\omega^2 A^2 sin^2(\omega t +\phi) dt $

$ \displaystyle U_{avg}= \frac{1}{4}m\omega^2 A^2 $

The average value of K.E. for complete cycle

$ \displaystyle K_{avg}= \frac{1}{T}\int_{0}^{T}K dt $

$ \displaystyle K_{avg}= \frac{1}{T}\int_{0}^{T}\frac{1}{2}m\omega^2 A^2 cos^2(\omega t +\phi)dt $

$ \displaystyle K_{avg}= \frac{1}{4}m\omega^2 A^2 $

Thus average values of K.E. and P.E. of harmonic oscillator are equal and each equal to half of the total energy

Kaverage = Uaverage

$ \displaystyle \frac{1}{2}E = \frac{1}{4}m\omega^2 A^2 $

Exercise : The amplitude of an SHM is doubled. Find the corresponding change in

(a) time period

(b) maximum velocity

(c) maximum acceleration

(d) total energy

Illustration : A particle executes SHM .

(a) What fraction of total energy is kinetic and what fraction is potential when displacement is one half of the amplitude;

(b) At what displacement the kinetic and potential energies are same?

Solution : (a)

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

For x = A/2

$ \displaystyle K = \frac{1}{2}m\omega^2 (A^2 – x^2) = \frac{1}{2}m\omega^2 \frac{3}{4}A^2 $

$ \displaystyle \frac{K}{E} = \frac{3}{4} $

$ \displaystyle U = \frac{1}{2}m\omega^2 x^2 = \frac{1}{2}m\omega^2 \frac{1}{4}A^2 $

$ \displaystyle \frac{U}{E} = \frac{1}{4} $

(b) K = U

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

=> x = 0.707A

Also Read :

→ Stable , Unstable & Neutral Equilibrium
→ S.H.M :Linear SHM & Angular SHM
→ Analytical Treatment to SHM
→ Kinetic Energy & Potential Energy & Total Energy in SHM
→ 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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