Waves and Fields (PHYS 195): Week 4 Spring 2019 v1.0 Intro: This week we finish our study of damped driven oscillators and resonance in these systems and start on our study of waves, beginning with transverse waves on a string. The first mid-term is this coming week during lab sections (Feb 21-22). Choose a lab section (Thursday or Friday) to take the midterm. Obviously, there will be not lab this week. Problem sets will not be due this week. Instead the problems on this guide are example mid-term questions. Reading: Friday: HRW 16.1-16.3 Monday: HRW 16.4 Wednesday: HRW 16.5-6 Physics Topics: Resonance The dependence of amplitude and phase of driving angular frequency ω Waves - equation of motion Transverse waves Phasors Math Topics: Partial derivatives Wave equation (in 1 dimension) Problems: Practice problems for the upcoming midterm. Solution hints are at the end of this guide. (1) Equations! (a) Write down the equation of motion for a simple harmonic oscillator (b) Circle the angular frequency in the equation. (c) Write the general solution for SHM (2) Sketch the amplitude as a function of driving frequency, for a damped, driven harmonic oscillator. (3) Please correct the following statement: If you drive a damped harmonic oscillator at a driving frequency other than ω o then at late times it oscillates at ω o but the amplitude of the response depends on how close the driving frequency is to the resonance frequency. (4) Using a simple pendulum you find the local acceleration of gravity. After careful data taking you find that the period is T = 2.7585 ± 0.0008 s and the length of the pendulum is l = 189.0 ± 0.2 cm. (a) What is your result for g? (b) What is the uncertainty δg? (c) Does it agree with g = 9.804 ± 0.005 m/s 2? 1
2 Please show all your work. (5) A mass m = 60.0 g hangs from a k = 3.30 N/m spring. (a) What is the angular frequency of oscillation? (b) You observe that the mass looses half its amplitude in 10.5 s. What is the general solution for the position of the mass? (c) When t = 0 the mass is at equilibrium and is moving upward at 1.20 m/s. Find the specific solution that describes the motion. (6) You pilot a spacecraft to a black hole with mass M and enter an orbit. At a radial distance from the planet r, your potential energy is ( U(r) = U o R ) R2 + a2 r r 2 where U o, R, and a are all constants and 0 < r <. Assume the spacecraft has a mass m. (a) Find the equilibrium position of the spacecraft. (b) Find the first three non-vanishing terms of the Taylor series around the stable equilibrium point. (c) Find k eff. (d) What is the angular frequency of small oscillations in the radial position of the spacecraft? This means that the spacecraft orbits the black hole at a radius that undergoes simple harmonic motion. (7) A small cuckoo clock has a pendulum 25 cm long with a mass of 11 g. The clock is powered by a 210 g weight which falls 2 m each day. The amplitude of the swing is 0.20 radians. What is the Q of the clock? (8) Many modern towers contain huge damped oscillator systems designed to oscillate at the same frequency as the buildings themselves. For instance the Taipei 101 tower has a 728 ton pendulum built into the 90-87th floors. You can view a video of the relative motion during an earthquake on this same web page. (a) Why are these damped oscillator systems built? (b) In the video the period of oscillation is about 7.1 s. Assuming a lightly damped simple pendulum, find the natural angular frequency. (c) Suppose that in 10 periods the amplitude of oscillation is reduced from the maximum of 1.4 m to 0.80 m. Find the effective damping coefficient b. (9) View the second video showing masses on springs. The display on the function generator is in Hz. (a) Describe what happens during the video. (b) The mass on the super-bouncy mass-on-a-spring is 10.0 g. What is the spring constant? (10) Describe three ways to find Q. Two ways we used on Guide 3, one way in lab. Lab: Mid-Term I! A look ahead... We move onto harmonics and sound next week. To read ahead see the end of Chapter 16 and the beginning of Chapter 17.
3 Problem Hints: (1) Equations! (a) (b) That s ω o. (c) d 2 x dt 2 + ω2 ox = 0. x(t) = x m cos(ω o t + ϕ) (2) See your Lorentzian curve, A 2 (f) vs. f, from lab. (3) The second ω o should be an ω - the driving angular frequency - since the damped, driven system oscillates at the driving frequency. Otherwise the statement is correct. (4) g (a) 9.8056 m/s 2 (b) 0.016 or 0.0118 so 0.02 or 0.01 depending on method (c) Yes (5) SHM (a) 7.42 1/s (b) y(t) = y m e 0.066t cos(7.42t + ϕ) (c) y(t) = 16.2 e 0.066t sin(7.42t) cm (6) SHM in orbits (a) r o = 2a 2 R (b) Taylor series... (c) (d) k eff = U o 8a 6 R 2 Uo ω = 8ma 6 R 2 (7) The Q of the clock is 71. (8) High DDHM: (a) When the tower sways it drives the damped mass-on-a-spring system. (b) The damping removes energy from the building s oscillation, reducing the amplitude of the swaying motion. This is more comfortable for people in the building and reduces strain on the building structure. Without the damping system one could build a less flexible structure to achieve the same end but this requires lots of costly steel. These systems save money. For instance, the system in the Citicorp building in NYC cost $ 1.5 million and is estimated to have saved $ 3.5-4 million in 2,800 tones of structural steel which would have been required to stiffen the structure. ω o = 2π 0.88 s1 T (c) Since A(t) = x m e βt we have Hence, A(t 0 ) = x m e βt0 = 1.4 m A(t 0 + 10T ) = x m e βt0+10t = 0.8 m A(t o + 10T ) A(t o ) = e β10t = 0.8 1.4 (1)
4 Hence β = ln(1.4/0.8)/(10t ). Using the relation between β and the damping coefficient b, 2m ln 1.75 2(728 1000) ln 1.75 b = = 1.15 10 4 1 10 4 kg s 1. 10T 10 7.1 You may have used a different definition of ton, which is fine. (9) Video 2 (a) The video shows a speaker driving a flexible bar supporting masses on springs. When the driving force is turned on at 2.60 Hz only one oscillator responds with large amplitude. It is in resonance. (b) Since the super bouncy mass is in resonance, then the driving angular frequency is the resonant angular frequency, or ω o for lightly damped systems, k ω = 2πf = ω o = m. So the spring constant is k = 4π 2 mf 2 = 4π 2 (10g)(2.6 Hz) 2.67 N/m where I assumed 3 sig figs. 2 or 4 are also fine. (10) Curve fit (as we did in lab) with ω o /2β, and with E stored ω o / E/ t.
5 Handy Relations General: τ = Iα I = r 2 dm F = du dx F B = ρgv The Taylor series of a function f(x) around x = x o is f(x) = f(x o ) + df dx (x x o ) + 1 d 2 f x=xo 2 dx 2 (x x o ) 2 + 1 d 3 f x=xo 3! dx 3 (x x o ) 3 +... x=xo Oscillations: For spring-like SHM ω o = The energy stored in an oscillator is F = kx k m, a simple pendulum ω o = g l, and a physical pendulum ω o = T = 2π ω o and ω = 2πf. E = 1 2 mω2 ox 2 m. A damped harmonic oscillator has the equation of motion The solution is Q = de dt E 1 ω o d 2 x dx + 2β dt2 dt + ω2 ox = 0. ω o 2β = mω o b and A(t) = x m e βt x(t) = x m e βt cos(ω d t + ϕ) with ω d = ω 2 o β 2 ω o. For a driven system at late times x(t) = A(ω) cos[ωt δ(ω)] with ( ) F o /m 2βω A(ω) = and δ(ω) = arctan [(ωo 2 ω 2 ) 2 + 4β 2 ω 2 ] 1/2 ωo 2 ω 2 where β = b/2m. The resonant angular frequency is ω R = ω 2 o 2β 2. ] k eff = d2 U dx 2 x=x o Uncertainty: Add independent uncertainties in quadrature. For addition and subtraction, add the uncertainties, z = x + y or z = x y then δz = δx 2 + δy 2 For multiplication and division then add the relative uncertainties, (δx z = xy or z = x/y then δz ) 2 z = + x ( ) 2 δy y mgh I.
6 For a power, multiply the relative uncertainty by the power, i.e. if z = x n then δz z = nδx x. In general for a calculated quantity q = q(x,..., z) then δq = ( q x δx ) 2 +... + ( ) 2 q x δz