Physics 325 Spring 2011 Homework 5

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

Ask a homework question - tutors are online