Science:Math Exam Resources/Courses/MATH215/December 2011/Question 04 (a)
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Question 04 (a) 

Suppose the motion of a springmass system is described by the following differential equation (a) Find all values of the parameter for which the system is overdamped or critically damped (that is the mass cannot pass its equilibrium position more than once, so there are no oscillatory solutions). 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
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Hint 

What are the roots of the characteristic equation as a function of . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Given the equation we can obtain the characteristic equation by substituting a solution and looking at the resulting polynomial. Doing this we get which has roots
Now we study the behaviour of the roots of the equation. Note that is necessary for physical damping.
Thus, there is overdamping for and underdamping for (and therefore critical damping if ). 