Physics 325 Spring 2011 Homework 5

# Physics 325 Spring 2011 Homework 5 - Homework 5A...

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Unformatted text preview: Homework 5A Physics 325 Spring 2011 Name: ____________________________ Due: Mar 16, 2011 by 9am (lecture or 325 box) 1. (10 points) This problem concerns the relationship between the average properties of the kinetic and potential energy for a simple harmonic oscillator. (a) Calculate the time averages of the kinetic and potential energies over one cycle and show that these are equal. Why is this a reasonable result given what you’ve learned previously in this course? (b) Calculate the spatial averages of the kinetic and potential energies. Compare these and discuss (a sketch of T ( x ) and U ( x ) might help). 2. (10 points) Calculate the average rate of energy loss dE dt (i.e. compute the time average over one cycle) for a lightly damped oscillator ( ! ! " 0 , where ! and ! 0 are the damping parameter and characteristic angular frequency in the absence of damping, respectively) in terms of m, ! , " 0 , A, t as defined in M&T Section 3.5. 3. (10 points) Show that x(t ) = ( A + Bt )e! " t is a solution to the case of critically damped motion by assuming a solution of the form x (t ) = y(t )e! " t and determining the function y(t ) . Homework 5B Physics 325 Spring 2011 Name: ____________________________ Due: Mar 16, 2011 by 9 am (lecture or 325 box) 4. (10 points) Show that, if a driven oscillator is only lightly damped and driven near resonance, that the Q of the system is approximately given by \$ ' Total Energy Q ! 2" # & % Energy loss during one period ) ( 5. (10 points) An undamped driven harmonic oscillator satisfies the equation of motion 2 m(d 2 x dt 2 + ! 0 x ) = F (t ) The driving force F (t ) = F0 sin(! t ) is switched on at t = 0 . (a) Find x (t ) for t > 0 for the initial conditions x = 0 and v = 0 at t = 0 (b) Find x (t ) for ! = ! 0 by taking the limit ! " ! 0 in your result for part a) [Hint: In part (a), look for a particular solution of the differential equation of the form x = A sin(! t ) and determine A . Add the solution of the homogeneous equation to this to obtain the general solution of the inhomogeneous equation. ...
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