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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.
F_(d)=2N
gamma =(2N)/((lm)/(s))
u^('')(t)+2U^(')(t)+5U(t)=0
(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^(2)e^(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.
Page 2 of 3
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 ut),the position of the object at time t. F=2N 8=2N 0L+2U'L+5U=0 1m/5
(b) Find the characteristic equation for the differential equation from part (a) by plugging in u = ert. Note: The characteristic equation was defined in the previous in-class worksheet. +2r+5e=
22r+S=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=-1. We'll talk about finding a general solution to this differential equation in a future worksheet.