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(27) The Pendulum and Damped and Forced Oscillations

(27) The Pendulum and Damped and Forced Oscillations -...

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Lecture 27 The Pendulum Damped and Forced Oscillations
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ACT: Energy oscillations A mass m = 5 kg oscillates at the end of a spring of constant k = 2000 N/ m. At t = 0, its acceleration is maximum. How long will it take before the potential energy reaches its next maximum? A. 0.31 s B. 0.16 s C. 0.08 s
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To get to the next peak in U, it takes half a period 1 2 0.16 s 2 2 T m k π π ϖ = = = At t = 0: Max a Max x Max U t x T U t
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The simple pendulum The simple pendulum L R c b s θ θ θ θ = = = sin cos b c s R R R A mass m is suspended at the end of a massless string of length L. Find the frequency of the oscillations for small displacements. Small θ ; ; b s c R θ θ θ sin cos 1 Small θ θ
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θ L The oscillations are rotations about point P. τ τ θ = = net sin g mgL θ θ - = 2 2 2 sin d mgL mL dt Newton’s second law: ( 29 τ α = net I T mg P θ θ + = 2 2 0 g d dt L SHM θ θ ϖ ϕ ϖ = + = 0 cos( ) wit h t g L Solution: θ θ - = 2 2 2 d mgL mL dt Small θ DEMO: Pendulum
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The physical pendulum The physical pendulum = a rigid body that oscillates about an axis (P).
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