I have a problem and I want to determine the friction coefficient using diffential equations i.e. solving the equation (this is from a Under-damped Oscillator)

$$ m\ddot x+c\dot x+kx=0. $$

Suppose I know the mass of the object, the initial distance from the origin, the constant of the spring, and the time in which the object is stopped. Can I solve my problem with this data? Also is there a better way to determine the friction coefficient using differential equations?

  • 1
    $\begingroup$ Does the object ever stop? Do you know how to solve the equation? $\endgroup$
    – G. Smith
    Commented Nov 21, 2018 at 3:33

1 Answer 1


Because it is a second order differential equation you always need two initial conditions to find the general solution. If you have the initial position and momentum, the value of $ k $ and $ m $ and the time at which it stops then you can find $ c $. The general solution to the equation is: $$ x_{(t)}=Ae^{-\frac{ct}{2}}\cos{\left(\sqrt{\frac{k}{m}}t + \phi\right)} .$$ The motion never really stops according to this model but the critical time is given by $ \tau=\frac{2}{c} $ so from there you could find the value of $ c $. I hope this helps you find your answer.

  • $\begingroup$ It's possible to calculate the kinetic coefficiet friction after this point? I'm not sure to do things like $cv=^?\mu mg$ $\endgroup$ Commented Nov 21, 2018 at 4:25
  • $\begingroup$ I think I misunderstood your question. It is important to note that the Coulomb model of friction and the one in the differential equation are very different. The one intue differential equation states that thefriction is proportional to the velocity (like in a fluid) but the Coulomb model states that it is proportional to the normal force. If you wanted to solve that problem you would have to consider the static and dinamic coefficent every time the mass started and stopped moving, which is a pain. $\endgroup$
    – Kirtpole
    Commented Nov 21, 2018 at 4:32
  • $\begingroup$ @Kirtpole most probably the dynamical friction would relate to the object's velocity. We perhaps can ignore the static as the mass moves then the equation becomes $m\ddot{ r}+ \mu N +kx =0$ $\mu = dynamic friction coefficient. $\endgroup$ Commented Nov 29, 2018 at 13:36

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