lecnotes1 - 1 1.1 Pendulum Free oscillator To introduce...

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1 Pendulum 1.1 Free oscillator To introduce dynamical systems, we begin with one of the simplest: a free oscillator . SpeciFcally, we consider an unforced, undamped pendulum. The arc length (displacement) between the pendulum’s current position and rest position ( β = 0) is s = Therefore ˙ s = ˙ ¨ ¨ s = ±rom Newton’s 2nd law, ¨ F = mlβ The restoring force is given by mg sin β . (It acts in the direction opposite to sgn( β )). Thus ¨ F = mlβ = mg sin β or d 2 β g + sin β = 0 . d t 2 l l mg mg sin θ θ Our pendulum equation is idealized: it assumes, e.g., a point mass, a rigid geometry, and most importantly, no friction . 7
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± ² ± ³ The equation is nonlinear, because of the sin β term. Thus the equation is not easily solved. However for small β 1 we have sin β β . Then d 2 β g β d t 2 = l whose solution is g β = β 0 cos t + θ l or β = β 0 cos( γt + θ ) where the angular frequency is g γ = , l the period is l T = 2 α , g and β 0 and θ come from the initial conditions.
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This note was uploaded on 02/24/2012 for the course MECHANICAL 12.006J taught by Professor Danielrothman during the Fall '06 term at MIT.

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lecnotes1 - 1 1.1 Pendulum Free oscillator To introduce...

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