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# s4 - note that INCREASING ω increases D(say ω = 21 gives...

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Phys 263/Amath 261 Assignment 4 SOLUTIONS 1. (a) Look at the equilibrium position, and equate mg = kx k = mg x = 1000 × 9 . 8 0 . 1 = 98000 N / m (1) (b) The undamped oscillation period is T = 2 π ω 0 = 2 π r m k = 0 . 63s (2) (c) At critical damping, γ = ω 0 , b = 2 = 1 . 98 × 10 4 Kg / s (3) (d) Since the mass has changed, we calculate ω 0 0 = r k m = r 98000 1000 + 3000 = 4 . 95 rad / s (4) and γ 0 = b 2 m = 2 . 48 rad / s (5) Since γ 0 is smaller than ω 0 0 , the motion is underdamped. (e) The oscillation period of the loaded van is now T 0 = 2 π p ω 0 2 0 - γ 0 2 = 1 . 47s (6) (f) When resonance occurs, the period of the loaded van is T 0 = 2 π ω r = 2 π p ω 0 2 0 - 2 γ 0 2 = 1 . 8 s (7) And we have a wavelength, with which we can ﬁnd the speed v = λ T 0 = 20 1 . 8 = 11 . 1 m / s (8) 2. This can be solved either by assuming the steady-state solution, or working out the equations of motion from ﬁrst principles. Assuming the steady-state solution, we can use the amplitude given in class: D = F 0 /m p ( ω 2 o - ω 2 ) 2 + 4 γ 2 ω 2 (9) 1

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and take the limit of no damping, γ = 0. Then D = F 0 /m | ω 2 o - ω 2 | = 18 | 1000 - ω 2 | (10) And require that - 0 . 03 D 0 . 03. Solving the equation 0 . 03 = 18 1000 - ω 2 (11) gives two solutions: ω = 20 and ω = - 20. The ﬁrst one is the only physical one. Also
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Unformatted text preview: note that INCREASING ω increases, D , (say ω = 21 gives D = 0 . 0322). So ω ≤ 20. Also note that-. 03 = 18 1000-ω 2 (12) gives two solutions: ω = 40 and ω =-40. Keeping the second one and noting that DECREASING ω decreases (makes more negative) D , requires ω ≥ 40. So the range of allowed frequencies are ω ≤ 20 and ω ≥ 40 . (13) 3. The following Lissajous curves were plotted on Maple with the included commands: (a) plot([cos(t-(1/5)*Pi), cos(2*t), t = -2*Pi . . 2*Pi]) (b) plot([cos(5*t-Pi), cos(4*t-(1/2)*Pi), t = -2*Pi . . 2*Pi]) (c) plot([cos(10*t-(1/2)*Pi), cos(11*t-Pi), t = -2*Pi . . 2*Pi]) (d) plot([cos(Pi*t), cos(t), t = -20000*Pi . . 20000*Pi]) Note in this last case, ω x /ω y is not a rational fraction, eﬀectively ﬁlling the square. 2 3...
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s4 - note that INCREASING ω increases D(say ω = 21 gives...

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