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Unformatted text preview: Physics 315: Oscillations and Waves Midterm 1: Solutions 1. (a) ω is the characteristic angular oscillation frequency of the system. (b) The characteristic frequency in cycles per second is f = ω/ 2 π . (c) x ( t ) = x cos( ω t ). (d) Now, d E dt = 2 ˙ x ¨ x + 2 ω 2 x ˙ x, or d E dt = 2 ˙ x ( ¨ x + ω 2 x ) . However, ¨ x + ω 2 x = 0. Hence, d E /dt = 0. E is twice the energy per unit mass of the system. 2. (a) ν parameterizes the amount of damping in the system. (b) If x ( t ) = A sin( ω 1 t ) e γ t then ˙ x ( t ) = A ω 1 cos( ω 1 t ) e γ t + A γ sin( ω 1 t ) e γ t , and ¨ x ( t ) = A ω 2 1 sin( ω 1 t ) e γ t +2 A ω 1 γ cos( ω 1 t ) e γ t + A γ 2 sin( ω 1 t ) e γ t . Hence, substitution into the damped harmonic oscillator equation yields 0 = bracketleftbig ω 2 ω 2 1 + γ ν + γ 2 bracketrightbig sin( ω 1 t ) + [2 γ ω 1 + ν ω 1 ] cos( ω 1 t ) ....
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 Spring '08
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 Physics, Normal mode, Eqs., damped harmonic oscillator, antiphase

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