Average Value of P.E. and K.E.
Category : JEE Main & Advanced
The average value of potential energy for complete cycle is given by
\[{{U}_{average}}=\frac{1}{T}\int_{\,0}^{\,T}{U\,dt}=\frac{1}{T}\int_{\,0}^{\,T}{\frac{1}{2}m\,{{\omega }^{2}}{{a}^{2}}{{\sin }^{2}}(\omega \,t+\varphi )}\]\[=\frac{1}{4}m{{\omega }^{2}}{{a}^{2}}\]
The average value of kinetic energy for complete cycle \[{{K}_{average}}=\frac{1}{T}\int_{\,0}^{\,T}{K\,dt}\]\[=\frac{1}{T}\int_{\,0}^{\,T}{\frac{1}{2}m\,{{\omega }^{2}}{{a}^{2}}{{\cos }^{2}}\omega \,t\,dt}=\frac{1}{4}m{{\omega }^{2}}{{a}^{2}}\]
Thus average values of kinetic energy and potential energy of harmonic oscillator are equal and each equal to half of the total energy \[{{K}_{average}}={{U}_{average}}\]\[=\frac{1}{2}E=\frac{1}{4}m{{\omega }^{2}}{{a}^{2}}\].
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