Re: [asa] ICR's GENE project

Burgy -

You're on the right track. When the pendulum is at an angle w with the vertical, the component of the gravitational force perpendicular to the pendulum is mgsinw & the torque on the bob mgLsinw if L is the length. Equating this to the change in angular momenum -d(mL^2dw/dt)/dt gives -Ld^2w/dt^2 = gsinw or w'' + (g/L)sinw = 0. If
w << 1 (then sinw ~ w & w'' + (g/L) = 0. This is the equation for a harmonic oscillator with period 2pi*sqrt(L/g) independent of amplitude.

If w isn't small then the solutions are elliptic functions. The period in general is 4K[sin(A/2)]*sqrt(L/g) where K is the complete elliptic integral of the 1st kind & A is the amplitude. When A is 90 degrees (which in practice requires a rigid "massless" rod rather than a string) this works out to ~7.4sqrt(L/g), more than twice the small amplitude period.

Shalom
George
http://web.raex.com/~gmurphy/
  ----- Original Message -----
  From: Carol or John Burgeson
  To: gmurphy@raex.com ; asa@calvin.edu
  Sent: Saturday, June 30, 2007 12:26 PM
  Subject: Re: [asa] ICR's GENE project

>>Almost without fail students "confirmed" Galileo's conclusion - in spite of the fact that it's wrong, & that period is independent of amplitude only for small oscillations>>

  That's one I never heard. Tell me more. I would guess that the period actually changes continuously as the amplitude increases; this due to the direction of the force vector continually changing.

  Fascinating. I wish D. Lytle Wiggins had taught us that in my 1947 high school physics class!

  Burgy

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Received on Sat Jun 30 20:03:19 2007