Potential Energy

• Length

• hence dl-dx = (1/2) (dy/dx)2 dx

• dU = (1/2) F (dy/dx)2 dx potential energy of element dx

• y(x,t)= ym sin( kx- ?t)

• dy/dx= ym k cos(kx - ? t) keeping t fixed!

• Since F=?v2 = ??2/k2 we find

• dU=(1/2) ?dx ?2ym 2cos2(kx- ? t)

• dK=(1/2) ?dx ? 2ym 2cos2(kx- ? t)

• dE= ??2ym 2cos2(kx- ?t) dx

• average of cos2 over one period is 1/2

• dEav= (1/2) ? ? 2ym 2 dx

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