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Simple harmonic motion problem

  1. Nov 30, 2004 #1
    In a block-spring system with a block of mass m and a spring of spring constant k, prove that the angular velocity ω of the block = (k/m)^(1/2).

    I can prove this easily in the following manner:

    F = ma Newton's law
    F = -kx Hooke's law
    a = -ω^2x
    ma = -kx = -mω^2x
    k = mω^2
    ω^2 = (k/m)
    ω = (k/m)^(1/2)

    But when I try to prove it using calculus (as my teacher instructed me to do) something goes wrong:

    F = ma
    F = -kx
    X = Acos(ωt + φ) SHM
    F = ma = -kx = -kAcos(ωt + φ)

    I integrated to get:

    mv = -kωAsin(ωt + φ)

    Did I do something wrong here? I kept integrating (so that there was a ω^2 term on the right side) and substitued for x = Acos(ωt + φ) and cancelled out x; but then I got ω = (m/k)^1/2. I don't understand why---is there a flaw in the math somewhere? I think I can probably do the proof by simply integrating the SHM equation and substituting it for acceleration, but I'd like to know what I did wrong above.

    Thanks for any help :-)
  2. jcsd
  3. Nov 30, 2004 #2
    You messed up your integral. What you've found is the derivative without the negative sign.

    What's easier is to realize that a = [itex] \Ddot{x}[/itex], and then just differentiate x twice with respect to time and plug it in for a, then compare terms.

  4. Nov 30, 2004 #3
    Thanks! I didn't realize I was integrating with the chain rule :-) I have to do integration by substitution.
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