Unformatted text preview: Oscillator Homework Problems Michael Fowler 3/20/07 1. Dimensional exercises: use dimensio ns to find a characterist ic time for an undamped simple harmo nic oscillator, and a pendulum. Why does the dimensio nal argument work for any init ial displacement of the oscillator, but only small swings of the pendulum? What possible characteristic times can be found dimensionally for a damped oscillator? Explain the physical significance of these times for a heavily damped oscillator, and a light ly damped oscillator. 2. Open the damped oscillator spreadsheet. Let’s first examine damped mot ion without the spring. Set m = 1, k = 0, b = 3, xinit = 0, vinit = 3. (a) How does the curve relate to the dimensio nally derived time(s) for a damped oscillator? (b) Write down the equation for this damped k = 0 “oscillator”. (Of course, this won’t oscillate!). Put dx/dt = v, to get a firstorder equation for v. Solve it, and see if your so lut ion agrees with the spreadsheet curve. (c) Now you’ve found v(t), you know dx/dt. Write down and so lve the equat ion for x(t), and check that it agrees with the spreadsheet. (d) Bring back the spring: set k = 1. Does this significant ly change the init ial shooting upwards of the curve? What, then, are the important terms in the equation for that part of the motion? (e) Look at the very top of the curve, the maximum value of x: what are the important terms in the equation in that neighborhood? (f) For longer times, which terms in the equation dominate. Drop the least important term, and solve the remaining equat ion. Then check to see if this is a good approximat ion or not. 3. Open the damped oscillator spreadsheet. Fix the constants to give the crit ical damping curve. Then lower the damping b unt il you can detect the oscillator going past the origin. (a) Roughly, by what percentage do you need to lower b to see this? (b) Suppose in building a model for a shock absorber you were willing to let the downward swing be as much as 5% of the original upward displacement, and you take m = 1, k = 1 for the model, what would be the value of b? 4. Open the damped dr iven oscillator spreadsheet and put k = 1, b = 0.1, omega = 1.25, delta_t = 0.055. (a) What’s go ing on at the beginning? It might help to set b = 0 temporarily to get some insight. 2 (b) Note that the solut ion settles down to a steady state. Does the time to settle down depend on the init ial condit io ns? Change them and find out. Set xinit large, for example. Can you arrange the init ial condit io ns so that the steady so lut ion takes over immediately? How would you do that? 5. (a) Open the pendulum spreadsheet. See how the period varies wit h the amplitude: it’s init ially set at 0.1 radian. Try 1 radian, 2 radians, 3 radians. (b) In the pendulum spreadsheet, set the init ial angle theta_init =0, then try the init ial angular velocit y omega_init = 4, 5, 6, 7. Interpret your result. How can you make the pendulum period very lo ng? 6. An unpowered streetcar is accelerat ing under gravit y down a ten degree slope. Neglect frict ion and air resistance, assume the acceleration is the same as a smooth block sliding down a frict ionless surface. A pendulum o f length l is hung fro m the ceiling inside. (a) If the pendulum is at rest, what is the direct ion of the string? ? (b) What is the period of oscillat ion of the pendulum? 7. Some future civilizat ion bores a tunnel vert ically down, through the center of the earth, emerging at the opposite point on the globe. The air is pumped out of the tunnel, leaving a vacuum. Passenger pods are dropped into the tunnel. Assume any frict ion to be negligible. (a) How long does a oneway trip to the other end of the tunnel take? (b) How does that compare wit h a satellite in low orbit getting from one end to the other? (c) Another straightline tunnel is bored from New York to Los Angeles. The air is removed, trains run on frictionless magnet ic rails. How long does the oneway trip take? 8. A 1 kg mass rests on a spring. A gent le downward pulse causes vertical oscillat ions at 5Hz. (a) Suppose a balloon is attached to the top of the mass. The balloon has a mass of 0.05kg, but feels a buo yancy force able to lift 0.55 kg. How does this affect the period of the oscillat ion? (b) What would be the period of the mass + spring (no balloon) on the Moon? (gMoon = 2 m/s2 .) 3 9. A spring is hanging vert ically at rest. A mass held in the hand is gent ly attached to the end of the spring, then released. The system oscillates, the maximum downward distance being 3 cm below the original posit ion. What is the period of oscillat ion? 10. Let us represent a ship weighing 20,000 tons (1 ton = 1,000 kg) by a rectangular parallelepiped, 150 m long, 30 m across, 20 m deep. Show that in vertical mot ion, this ship behaves as a simple harmo nic oscillator, and find the period. 11. A flat horizontal plate driven fro m below oscillates vertically with an amplitude o f 1 mm. Some sand (of negligible mass) is sprinkled on the plate. The frequency o f the oscillator is gradually increased fro m zero. At what frequency will the sand lose contact with the plate? At what point in the cycle will this happen? 12. (a) Prove that for a light ly damped oscillator, the change in frequency caused by the damping is approximately w / 8Q 2 . 0 (b) If damping causes a 1% decrease in the frequency of an oscillator, what is its Q value? Over how many cycles does the energy drop by 1/e? Over how many cycles does the energy drop by ½? 13. A light ly damped oscillator has mass m, spring constant k and damping factor b. (a) Prove that at any instant of time the rate of loss of energy is bv2 joules/sec., v being the instantaneous velocit y. (b) Assuming the change in amplitude in a single cycle is negligible, what is the average value o f v2 over the cycle compared wit h the maximum value of v2 ? (c) If he energy loss in one second is small, show that it is well approximated by
E ( t = 1) = E ( t = 0 ) (1  b ´ / m ) , and deduce that for long times the energy decays as e  bt / m . 1
14. A light ly damped dr iven oscillator exhibits a strong resonance at frequency w . Prove that at 0 resonance, the total energy in the oscillator for a given driving force is proportional to Q2 . 4 15. An old but precisely made pendulum clock keeps time within one second a day in Charlottesville. The proud owner takes it to a new apartment in Wintergreen, about 3000 feet above Charlottesville in alt itude. If the clock is not adjusted, how many seconds a day will it gain or lose? (Assume the new room location is kept at the same temperature as the earlier place.) 16. The bob of a pendulum is a uniform disk o f radius 4 cm , attached to the end of a very light rod, so that the center of the bob is a distance 30 cm fro m the support axle (which would be perpendicular to the page). Axle perpendicular to page q (a) Find the mo ment of inert ia of the bob about the axle (you’ll need to use the Parallel Axis Theorem). (b) For small oscillat ions, what percentage error arises in using the simple pointmass approximat ion? 17. We proved in the lecture that the steadystate solut ion for the damped oscillator driven by a force
F ( t ) = F0 cos wt
is
x ( t ) = A cos (wt  q ) , where A = F 0 2
m 2 (w0  w 2 ) 2 + (bw ) 2 tan q = , bw . 2
m ( w0  w 2 ) (a) Prove that the total energy in the oscillator, kinetic + potential, usually varies through the cycle. Explain why, by co mparing the rate of working of the dr iving force and that of the damping force. Show that the variat ion disappears at resonance. (b) Prove that at the resonant frequency, the energy in the oscillator is 2 Q F0 2 Er esonance =
. 2 2 mw0 5 (c) Prove that the power input (rate of working) of the driver at resonance is Q F0 2 Pr sonance = . e
2 mw0 (d) The power input will drop to half on varying w away fro m resonance when the deno minator 2
2 m 2 (w0  w 2 ) 2 + (bw ) doubles. Assume Q is large, so you can replace bw by bw over this 0
range, and conclude the power input is halved at w  w0 @ ±w0 / 2Q. Sketch very roughly the power input as a funct ion of driving frequency for a large Q, then for double that Q on the same sketch. ...
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 Fall '07
 MichaelFowler
 Work, damped oscillator

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