F mg sin mg substitute x l or xl mg d2 x f

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Unformatted text preview: object suspended from the end of a light (negligible) weight cord that doesn’t stretch. F = −mg sin θ ≈ −mg θ substitute x = lθ or θ = x/l mg d2 x F ≈− x=m 2 l dt θ = θmax cos (ω t + φ) ω = f = T = ￿ g l ￿ ω 1 g = 2π 2π l ￿ 1 g = 2π f l The Physical Pendulum and the Torsion Pendulum τ = −mgh sin θ ￿ d2 θ τ = Iα = I 2 dt d2 θ I 2 = −mgh sin θ dt d2 θ I 2 + mgh sin θ = 0 dt For small angular amplitudes ￿ ￿ 2 dθ + 2 dt mgh I θ=0 θ = θmax cos (ω t + φ) ￿ 2π I T= = 2π ω mgh Damped Harmonic Mo/on •  In nature usually oscilla/ons are damped. The damping force depends on the speed of the oscilla/ng object. Fdamping = ma = −bv −kx − bv d2 x dx ⇒ m 2 +b + kx = 0 dt dt •  The solu/on of this second order differen/al equa/on is: x = Ae−γ t cos ω ￿ t b γ= 2m ω= ￿ ￿ k b2 − m 4m2 •  Under ­damped (A) b2 < 4mk •  Over ­damped (C) b2 >> 4mk •  Cri/cal damping (B) b2 = 4mk...
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This note was uploaded on 02/03/2014 for the course PHYSICS 1061 taught by Professor Tsankov during the Fall '09 term at Temple.

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