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Spring-Mass Equations without Damping
Due: Fri, Feb 16, 2024
3. Suppose we have an object with a mass of 1kg at the end of a spring whose spring coefficient is k = 5.
Also suppose the damping force measures 2N at a speed of 1 m/s, and there is no external force.
(a) Set up a 2nd order ODE for u(t), the position of the object at time t.
$u''(t) + 2u'(t) + 5u(t) = 0$
F_\beta = 2N
$\gamma = \frac{2N}{1m/s}$
(b) Find the characteristic equation for the differential equation from part (a) by plugging in u = e^{rt}.
Note: The characteristic equation was defined in the previous in-class worksheet.
r^2e^{rt} + 2re^{rt} + 5e^{rt} = 0
r^2 + 2r + 5 = 0
(c) What are the roots of the characteristic equation? That is, what values of r will work here?
Hint: Your answers should include i = \sqrt{-1}.
We'll talk about finding a general solution to this differential equation in a future worksheet.