Consider a simple harmonic oscillator. Calculate the time averages of the kinetic and potential energies over one cycle, and show that these quantities are equal. Next, calculate the space averages of the kinetic and potential energies.(adsbygoogle = window.adsbygoogle || []).push({});

I'm completely confused about these terms. Time average? Space average?

I do know that, with SHM, the equation of motion is -kx=ma, and w(omega)^2=k/m. The equation of motion becomes a+(w^2)x=0. The solution for this equation is x(t)=Asin(wt-delta). I used this x value for kinetic energy, (1/2)mv^2.

x(t)=Asin(wt-d)

v(t)=wAcos(wt-d)

T=(1/2)mv^2:

(1/2)m(w^2)(A^2)cos^2(wt-d)

Sub in w^2=k/m,

T=(1/2)k(A^2)cos^2(wt-d)

For potential energy,

U=(1/2)kx^2

U=(1/2)k(A^2)sin^2(wt-d)

Then, I found total energy, E=T+U.

E=(1/2)kA^2(cos^2(wt-d)+sin^2(wt-d))

E=(1/2)kA^2

The question cites that the average is over one cycle, which makes me think I should incorporate period, or frequency. Period=2pi*square root (m/k). I don't know where I'm going with this. I might have just done a bunch of superfluous calculations.

The answer for the time average is T=U=(m*A^2*w^2)/4. And the space average is U=(1/2)T=(m*A^2*w^2)/6. Actually, if someone could even explain what the question is asking for, it would be a great help.

Thank you so much!!

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# Homework Help: Harmonic Oscillation Energy

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