Calculating Energy Averages & Losses in Harmonic Oscillators, Assignments of Mechanics

Download Calculating Energy Averages & Losses in Harmonic Oscillators and more Assignments Mechanics in PDF only on Docsity! PHYS 419: Classical Mechanics, Assignment 7 Due 10/29/07 1. Taylor: Problem 5.4. 2. Consider a simple harmonic oscillator with period τ . Let 〈f〉 denote the time average of f(t), i.e., the value of f(t) averaged over one complete cycle: 〈f〉 = 1 τ ∫ τ 0 f(t)dt. (a) Calculate 〈T 〉 and 〈U〉, where T and U are the kinetic and potential energy, respectively, and show that E = 2〈T 〉 = 2〈U〉, where E is the total energy. [Hint: Start by proving the very useful result: 〈sin2(ωt − δ)〉 = 〈cos2(ωt − δ)〉 = 1 2 by considering the integral of sin2 θ + cos2 θ]. Discuss the relation between the averaged and maximum values of T and U . (b) Calculate the space averages of T and U , defined as f̄ = 1 2X ∫ X −X f(x)dx, where [−X, X] is the range of the motion. Relate the two quantites to the total energy and interpret them physically. 3. Taylor: Problem 5.18. 4. Taylor: Problem 5.25. (d) In another oscillator, the amplitude drops by a factor of e after four periods. Find the ratios β/ω0 and ω1/ω0. Discuss how the oscillatory character of the motion depends on these ratios. 5. Taylor: Problem 5.27. 6. Consider an underdamped harmonic oscillator. (a) Find an expression for the energy of an underdamped harmonic oscillator as a function of time and then, by calculating its derivative, an expression for the rate of energy loss due to dissipation. (b) Make an approximate plot of the latter expression assuming the damping param- eter β equal about 0.1 of the natural frequency ω0. Interpret the minima and maxima appearing on the plot. 1