# How is damping force related to frictional force?

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?

• Does the object ever stop? Do you know how to solve the equation? Commented Nov 21, 2018 at 3:33

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.
• It's possible to calculate the kinetic coefficiet friction after this point? I'm not sure to do things like $cv=^?\mu mg$ Commented Nov 21, 2018 at 4:25
• @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. Commented Nov 29, 2018 at 13:36