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Simple harmonic motion equations derivation?

  1. Nov 23, 2013 #1
    Well I was going through class lecture notes and my professor wrote this

    When x = A(the maximum value), v=0: E=1/2kA^2
    When v = wA, x=0: E=1/2mw^2A^2

    where w = omega, A = amplitude, k = spring constant, m = mass, v = velocity

    and apparently both equations are equal, i would like to know how and why
    please and thank you
     
  2. jcsd
  3. Nov 23, 2013 #2
    If there is no friction or dissipation,kinetic and potential energy are alternately transformed into each other in SHM,but the total mechanical energy E=K+U is conserved.

    Consider a spring with one end attached to the wall and other end having a block,going through its cycle of oscillation.

    Total energy E = (1/2)mv2 + (1/2)kx2

    At any position x, E = (1/2)mvx2 + (1/2)kx2

    At the extreme position ,i.e at x=A the block has only potential energy ,so E = (1/2)kA2

    At the equilibrium position ,at x=0 the block has only kinetic energy ,so E = (1/2)mvmax2

    Now we know,vmax = ωA ,so at x=0 E=(1/2)mω2A2
     
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