Physics 326 Homework #2 due in course homework box by Wed, 12:25 pmAll solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points for poorly written solutions or incorrect reasoning. Answers given without explanation will not be graded: NoWork = No Points. However you may always use any relation on the 1DMath, 3DMath or exam formula sheets or derived in lecture / discussion. Write your NAME and DISCUSSION SECTION on your solutions.

Problem 1 : Resonance the Q Factor, Part 1 adapted from Taylor 5.41We know that if the driving frequency is varied, the maximum response-squared (A2) of a damped oscillator that is driven with a sinusoidal force, f0 cos(t) , occurs at 0 for a weakly damped oscillator ( 0 ). As you know, 0 is the oscillators natural frequency = the frequency at which it would oscillate if it was undamped and undriven.

(a) Show that the peak value of the oscillators response (i.e. the amplitude at resonance) is Amax f0

20.

(b) Show that A2 note the square is equal to half its maximum value when , so that the full width of the response curve A2() at half maximum is just 2. [Taylors Hint: Be careful with your approximations. For instance, its fine to say +0 20 but you certainly must not say 0 0 .]Jargon note: FWHM is a standard acronym for full width at half maximum, which is a very standard measure of the width of a peak. (c) The Q factor of a peak is a measure of its sharpness and is defined as Q peak / FWHM ; yourwork in this question has shown that Q =0 / 2 for our damped, driven linear oscillator. If you stare at the definition of Q you can see that is a dimensionless number that indicates how close to the peak (resonant) frequency you must be to see the effects of the often-dramatic jump in response that occurs at resonance. Suppose you had an oscillating system with a resonant frequency of 200 Hz and a resonant response of Amax2 . What range of driving frequencies would produce a big response of A2 (Amax2 / 2) if the Q factor was 10? What if Q was 100?

Problem 2 : Resonance the Q Factor, Part 2 adapted from Taylor 5.44You probably wondered why the previous problem was concerned with the FWHM of the square of the oscillator amplitude. The reason is energy. The x in the equation x + 2 x +0

2x = f (t) denotes the oscillators state at any given moment, and A is the amplitude of that oscillating state but the energies involved in the system are all proportional to the squares of x and its derivatives. And so we come to a seconddefinition of the Q factor: it is also a measure of energy stored over energy lost/cycle. In symbols, this definition is Q = 2E / Edis where the subscript dis means the dissipated energy that is lost from the system during each oscillation period. Lets explore this definition for a mechanical oscillator.Consider the motion of a driven damped oscillator after any transients have died out, and suppose that it is being driven close to resonance, so you can set =0 .

(a) Show that the oscillators total energy = kinetic + potential is E = 12 m 2A2 .

(b) Show that the energy Edis dissipated during one cycle by the damping force is 2mA2 .

(c) Calculate the Q-factor from the energy-based definition, Q = 2E / Edis , and compare it to the result you obtained in problem 1 from the sharpness definition, Q peak / FWHM .

Problem 3: Car on a Bumpy Road Taylor 5.43When a car drives along a washboard road, the regular bumps cause the wheels to oscillatoe on the springs. (What actually oscillates is each axle assembly, comprising the axle and its two wheels.) Find the speed of my car at which this oscillation resonates, given the following information:(a) When four 80-kg men climb into my car, the body sinks by a couple of centimeters. Use this to estimate the spring constant k of each of the four springs. (b) If an axle assembly (axle plus two wheels) has total mass 50kg, what is the natural frequency of the assembly oscillating on its two springs? (c) If the bumps on a road are 80 cm apart, at about what speed would these oscillations go into resonance?

Problem 4: Two Springs, Two Dimensions Taylor 5.18The mass shown in the figure at right is resting on a frictionless horizontal table in the xy plane. Each of the two identical springs has force constant k and unstretched length l0. At equilibrium, the mass rests at the origin, with the distances a are not necessarily equal to l0 (i..e. at the equilibrium position shown, the identical springs have the same length a, but may already be stretched or compressed). (a) Show that when the mass moves to a position (x, y) with x and y small compared to the other distance scales in the problem (a and l0), the potential energy has the leading-order form U 12 kxx2 + kyy2( ) , where kx and ky are constants whose values you must determine. (b) Show that if a < l0, the equilibrium at the origin is unstable, and explain why in a few words that clarify what happens in physical terms.

Problem 5: Dont Break the Spring!A block of mass 2 kg is suspended from a fixed support by a spring of strength (= spring constant) 2000Nm1. The block is subject to the vertical driving force 36 cos(t) N, with a positive value indicating force in the downward direction (= same direction as gravity). No spring is infinitely sturdy, and this one will break if it is stretched more than 4cm beyond its unstretched length. (To be 100% clear: the unstretched length is the springs length when no forces are acting on it at all, e.g. when it is suspended without any mass attached.) Using g = 10m/s2, find the range(s) of driving frequency, , that can be safely applied to this system without breaking the spring.Procedure: This problem differs from the driven oscillator we solved in lecture in two ways: it involves gravity, and it has no damping force. To solve it, go back to the beginning: set up the equation of motion, obtain its particular solution (ignore the transient solutions to the homogeneous equation), then figure out the safe range(s) of driving frequency . For your coordinate system, I suggest you define the positive-x direction to point downward; Ill let you choose where to put the origin. Finally, no, I didnt forget to specify the unstretched length of the spring.