Show: a) that the motion equation's solution, x = asin wt:md2x&nbsp;dt&nbsp;2˙=−kx˙&nbsp;of the harmonic oscillator.&nbsp;b) that&nbsp;w=k{m}&nbsp;c) that the momentum is given by p =&nbsp;mwAcos⁡wt&nbsp;d) the particle is stationary if and only if x = A&nbsp;(To get the time when x = A, use the formula x = A sinwt. Then, use this time in an expression for the velocity.)&nbsp;e) that the energy of a harmonic oscillator is&nbsp;12kA2

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Show: a) that the motion equation&#039;s solution, x = asin wt:md2x&amp;nbsp;dt&amp;nbsp;2˙=−kx˙&amp;nbsp;of the harmonic oscillator.&amp;nbsp;b) that&amp;nbsp;w=k{m}&amp;nbsp;c) that the momentum is given by p =&amp;nbsp;mwAcos⁡wt&amp;nbsp;d) the particle is stationary if and only if x = A&amp;nbsp;(To get the time when x = A, use the formula x = A sinwt. Then, use this time in an expression for the velocity.)&amp;nbsp;e) that the energy of a harmonic oscillator is&amp;nbsp;12kA2

yagombyeR · Accepted Answer

a). The net force is given by F=−kx using Newton&#039;s second law F=m a where mis the mass and a is the acceleration F=m a=−k x acceleration (a) is second derivative ofx such that -k x = md2xdt2 solution to this differential equation is x=Acos⁡(wt)=Asin⁡wt b). To prove w=k/m substitute the values x=Acos⁡(wt+?) and d2xdt2=−Aw2cos⁡(wt) in −kx=md2xdt2−Akcos⁡(wt)=−Amw2cos⁡(wt) divide both sides by - Acos⁡(wt+)k=mw2w=km c). momentum is given by p=mv where v=dxdtp=mwAcos⁡wt e). Energy of the harmonic oscillator total E=P.E+K.E P.E=12kx2=12kA2cos⁡2(wt) K.E=12mv2=12m(dxdt)2=12mw2A2sin⁡2(wt) Total E=12kA2cos⁡2wt+12mw2A2sin⁡2wt=12kA2(cos⁡2wt+sin⁡2wt)=12kA2

Prove: a) that x = Asin wt is a solution of theequation of motion: m\dot \frac{d^{2}x}{dt^{2}}=-k\dot x of the harmonic oscillator. b) that w=\sqrt{k}

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