# Ce pt and ce pt the solution is the oscillating line

• 312

This preview shows page 107 - 109 out of 312 pages.

Ce - pt and - Ce - pt . The solution is the oscillating line between the two envelope curves. The enve- lope curves give the maximum amplitude of the oscillation at any given point in time. For example if you are bungee jumping, you are really inter- ested in computing the envelope curve so that you do not hit the concrete with your head. The phase shift γ just shifts the graph left or right but within the envelope curves (the envelope curves do not change if γ changes). Finally note that the angular pseudo-frequency (we do not call it a frequency since the solution is not really a periodic function) ω 1 becomes smaller when the damping c (and hence p ) becomes larger. This makes sense. When we change the damping just a little bit, we do not expect the behavior of the solution to change dramatically. If we keep making c larger, then at some point the solution should start looking like the solution for critical damping or overdamping, where no oscillation happens. So if c 2 approaches 4 km , we want ω 1 to approach 0. On the other hand when c becomes smaller, ω 1 approaches ω 0 ( ω 1 is always smaller than ω 0 ), and the solution looks more and more like the steady periodic motion of the undamped case. The envelope curves become flatter and flatter as c (and hence p ) goes to 0.
108 CHAPTER 2. HIGHER ORDER LINEAR ODES 2.4.4 Exercises Exercise 2.4.2 . Consider a mass and spring system with a mass m = 2 , spring constant k = 3 , and damping constant c = 1 . a) Set up and find the general solution of the system. b) Is the system underdamped, overdamped or critically damped? c) If the system is not critically damped, find a c that makes the system critically damped. Exercise 2.4.3 . Do for m = 3 , k = 12 , and c = 12 . Exercise 2.4.4 . Using the mks units (meters-kilograms-seconds), suppose you have a spring with spring constant 4 N / m . You want to use it to weigh items. Assume no friction. You place the mass on the spring and put it in motion. a) You count and find that the frequency is 0.8 Hz (cycles per second). What is the mass? b) Find a formula for the mass m given the frequency ω in Hz. Exercise 2.4.5 . Suppose we add possible friction to . Further, suppose you do not know the spring constant, but you have two reference weights 1 kg and 2 kg to calibrate your setup. You put each in motion on your spring and measure the frequency. For the 1 kg weight you measured 1.1 Hz, for the 2 kg weight you measured 0.8 Hz. a) Find k (spring constant) and c (damping constant). b) Find a formula for the mass in terms of the frequency in Hz. Note that there may be more than one possible mass for a given frequency. c) For an unknown object you measured 0.2 Hz, what is the mass of the object? Suppose that you know that the mass of the unknown object is more than a kilogram. Exercise 2.4.6 . Suppose you wish to measure the friction a mass of 0.1 kg experiences as it slides along a floor (you wish to find c ). You have a spring with spring constant k = 5 N / m . You take the spring, you attach it to the mass and fix it to a wall. Then you pull on the spring and let the mass go.