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# lecture35 - Damped Oscillations(Serway 15.6-15.7 Physics...

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Physics 1D03 - Lecture 35 1 Damped Oscillations (Serway 15.6-15.7)

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Physics 1D03 - Lecture 35 2 Simple Pendulum L θ T θ θ L g dt d - = 2 2 mg Recall, for a simple pendulum we have the following equation of motion: Which give us: L g = ϖ Hence: 2 2 2 4 π ϖ gT g L = = Application - measuring height - finding variations in g → underground resources or: 2 2 2 4 T L L g π ϖ = = -------------------------------------------------------------------------
Physics 1D03 - Lecture 35 3 t x x t SHM : x(t) = A cos ωt Motion continues indefinitely. Only conservative forces act, so the mechanical energy is constant. Damped oscillator : dissipative forces (friction, air resistance, etc. ) remove energy from the oscillator, and the amplitude decreases with time. SHM and Damping

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Physics 1D03 - Lecture 35 4 ) cos( ) ( 2 φ ϖ + = - t Ae t x t m b For weak damping (small b ), the solution is: f = - b v where b is a constant damping coefficient x t A damped oscillator has external nonconservative force(s) acting on the system. A common example is a force that is proportional to the velocity. eg: green water (weak damping) 2 2 dt x d m dt dx b kx = - - F=ma give:
Physics 1D03 - Lecture 35 5 Without damping: the angular frequency is 2 2 0 2 2 2 - = - = m b m b m k ϖ ϖ the amplitude gets smaller (decays exponentially) as time goes on: m k = 0 ϖ

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lecture35 - Damped Oscillations(Serway 15.6-15.7 Physics...

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