Physics I Oscillations and Waves. Questions

Transcription

1 Physics I Oscillations and Waves 1 Questions

2 Problems 1. An empty tin can floating vertically in water is disturbed so that it executes vertical oscillations. The can weighs 100 gm, and its height and base diameter are 20 and 10 cm respectively. [a.] Determine the period of the oscillations. [b.] How much mercury need one pour into the can to make the time period 1s? (0.227 Seconds, 1.73 cm ) 2. A SHO with ω 0 = 2 s 1 has initial displacement and velocity 0.1 m and 2.0 ms 1 respectively. [a.] At what distance from the equilibrium position does it come to rest momentarily? [b.] What are the rms. displacement and rms. velocity? What is the displacement at t = π/4 s? 3. A SHO with ω 0 = 3 s 1 has initial displacement and velocity 0.2 m and 2 ms 1 respectively. [a.] Expressing this as x(t) = Ãeiω 0t, determine à = a+ib from the initial conditions. [b.] Using à = Aeiφ, what are the amplitude A and phase φ for this oscillator? [c.] What are the initial position and velocity if the phase is increased by π/3? 4. A particle of mass m = 0.3 kg in the potential V (x) = 2e x2 /L 2 J (L = 0.1 m) is found to behave like a SHO for small displacements from equilibrium. Determine the period of this SHO. 5. Calculate the time average x 4 for the SHO x = A cos ωt.

3 6. An under-damped oscillator has a time period of 2s and the amplitude of oscillation goes down by 10% in one oscillation. [a.] What is the logarithmic decrement λ of the oscillator? [b.] Determine the damping coefficient β. [c.] What would be the time period of this oscillator if there was no damping? [d.] What should be β if the time period is to be increased to 4s? ([a.] [b.] s 1 [c.]2s [d.] 2.72s 1 7. Two identical under-damped oscillators have damping coefficient and angular frequency β and ω respectively. At t = 0 one oscillator is at rest with displacement a 0 while the other has velocity v 0 and is at the equilibrium position. What is the phase difference between these two oscillators. (π/2 tan 1 (β/ω)) 8. An LCR circuit has an inductance L = 1 mh, a capacitance C = 0.1µF and resistance R = 250Ω in series. The capacitor has a voltage 10 V at the instant t = 0 when the circuit is completed. What is the voltage across the capacitor after 10µs and 20µs? (7.64 V, 4.84 V )

4 Problems 9. Obtain solution (3.7) for critical damping as a limiting case (β ω 0 ) of overdamped solution (3.5). 10.Find out the conditions for the initial displacement x(0) and the initial velocity ẋ(0) at t = 0 such that an overdamped oscillator crosses the mean position once in a finite time. 11.A door-shutter has a spring which, in the absence of damping, shuts the door in 0.5s. The problem is that the door bangs with a speed 1m/s at the instant that it shuts. A damper with damping coefficient β is introduced to ensure that the door shuts gradually. What are the time required for the door to shut and the velocity of the door at the instant it shuts if β = 0.5π and β = 0.9π? Note that the spring is unstretched when the door is shut. (0.57s, m/s; 1.14s, m/s) 12.A highly damped oscillator with ω 0 = 2 s 1 and β = 10 4 s 1 is given an initial displacement of 2 m and left at rest. What is the oscillator s position at t = 2 s and t = 10 4 s? (2.00 m, m) 13.A critically damped oscillator with β = 2 s 1 is initially at x = 0 with velocity 6 m s 1. What is the furthest distance the oscillator moves from the origin? (1.10 m) 14.A critically damped oscillator is initially at x = 0 with velocity v 0. What is the ratio of the maximum kinetic energy to the maximum potential energy of this oscillator? (e 2 ) 15.An overdamped oscillator is initially at x = x 0. What initial velocity, v 0, should be given to the oscillator that it reaches the mean position (x=0) in the minimum possible time. 16.We have shown that the general solution, x(t), with two constants can describe the motion of damped oscillator satisfying given initial conditions. Show that there does not exist any other solution satisfying the same initial conditions.

5 Problems 17.An oscillator with ω 0 = 2π s 1 and negligible damping is driven by an external force F(t) = a cos ωt. By what percent do the amplitude of oscillation and the energy change if ω is changed from π s 1 to 3π/2s 1? (71.4%, 114%) 18.An oscillator with ω 0 = 10 4 s 1 and β = 1s 1 is driven by an external force F(t) = a cos ωt. [a.] Determine ω max where the power drawn by the oscillator is maximum? [b.] By what percent does the power fall if ω is changed by ω = 0.5s 1 from ω max?[c.] Consider β = 0.1s 1 instead of β = 1s 1. ( ([a.] 10 4 s 1, 33.3%, 96.2%) 19.A mildly damped oscillator driven by an external force is known to have a resonance at an angular frequency somewhere near ω = 1MHz with a quality factor of Further, for the force (in Newtons) F(t) = 10 cos(ωt) the amplitude of oscillations is 8.26mm at ω = 1.0 KHz and 1.0µm at 100 MHz. a. What is the spring constant of the oscillator? b. What is the natural frequency ω 0 of the oscillator? c. What is the FWHM? d. What is the phase difference between the force and the oscillations at ω = ω 0 + FWHM/2? f ω Show that, x(t) = (cosωt cosω ω 2 0 t), is a solution of the undamped forced system, ẍ + ω 0 x = f cos ωt, with initial conditions, x(0) = ẋ(0) = 0. Show that near resonance, ω ω 0, x(t) f 2ω 0 t sin ω 0 t, that is the amplitude of the oscillations grow linearly with time. Plot the solution near resonance. (Hint: Take ω = ω 0 ω and expand the solution taking ω 0.) 21.Find the driving frequencies corresponding to the half-maximum power points and hence find the FWHM for the power curve of Fig Show that the average power loss due to the resistance dissipation is equal to the average input power calculated in the expression (5.14). 23. (a) Evaluate average energies at frequencies, ω AmRes = ω0 2 2β 2 (at the amplitude resonance) and ω PoRes = ω 0 (at the power resonance). Show that they are equal and independent of ω 0. (b) Find the value of the forcing frequency, ω EnRes, for which the energy of the oscillator is maximum. (c) What is the value of the maximum energy? ((a) mf 2 /8β 2, (b) ωenres 2 = 2ω 0 ω 0 2 β 2 ω 2 0, ω AmRes < ω EnRes < ω PoRes, (c) mf 2 /16(ω 0 ω 2 0 β 2 ω β 2 ).)

6 24. A massless rigid rod of length l is hinged at one end on the wall. (see figure). A vertical spring of stiffness k is attached at a distance a from the hinge. A damper is fixed at a further distance of b from the spring providing a resistance proportional to the velocity of the attached point of the rod. Now a mass m(< 0.1ka 2 /gl) is plugged at the other end of the rod. Write down the condition for critical damping (treat all angular displacements small). If mass is displaced θ 0 from the horizontal, write down the subsequent motion of the mass for the above condition. 25. A critically damped oscillator has mass 1 kg and the spring constant equal to 4 N/m. It is forced with a periodic forcing F(t) = 2 cost cos 2t N. Write the steady state solution for the oscillator. Find the average power per cycle drawn from the forcing agent. 26. A horizontal spring with a stiffness constant 9 N/m is fixed on one end to a rigid wall. The other end of the spring is attached with a mass of k l m o a b 01 r kg resting on a frictionless horizontal table. At t = 0, when the spring -mass system is in equilibrium and is perpendicular to the wall, a force F(t) = 8 cos 5t N starts acting on the mass in a direction perpendicular to the wall. Plot the displacement of the mass from the equilibrium position between t = 0 and t = 2π neatly.