Chapter 5: Problem 32
Consider an underdamped oscillator (such as a mass on the end of a spring) that is released from rest at position \(x_{\mathrm{o}}\) at time \(t=0 .\) (a) Find the position \(x(t)\) at later times in the form $$x(t)=e^{-\beta t}\left[B_{1} \cos \left(\omega_{1} t\right)+B_{2} \sin \left(\omega_{1} t\right)\right]$$ That is, find \(B_{1}\) and \(B_{2}\) in terms of \(x_{\mathrm{o}}\). (b) Now show that if you let \(\beta\) approach the critical value \(\omega_{\mathrm{o}}\) your solution automatically yields the critical solution. (c) Using appropriate graphing software, plot the solution for \(0 \leq t \leq 20,\) with \(x_{\mathrm{o}}=1, \omega_{\mathrm{o}}=1,\) and \(\beta=0,0.02,0.1,0.3,\) and 1
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.