# Equations of motion of damped oscillations due to kinetic friction

phantomvommand
Take rightwards as positive.
There are 2 equations of motion, depending on whether ##\frac {dx} {dt} ## is positive or not.
The 2 equations are:
##m\ddot x = -kx \pm \mu mg##

Is this SHM?

Possible method to solve for equation of motion:
- Solve the 2nd ODE, although “joining” the equations when ##\dot x ## changes from positive to negative is not easy.

Gold Member
We may write the equation of motion as
$$m\ddot{x}=-kx- sgn(\dot{x})\mu mg$$
where sgn(a)=1 for a>0, 0 for a=0, -1 for a<0.

phantomvommand
phantomvommand
We may write the equation of motion as
$$m\ddot{x}=-kx- sgn(\dot{x})\mu mg$$
where sgn(a)=1 for a>0, 0 for a=0, -1 for a<0.
Thanks for this; do you then go on to solve the 2nd ODE?
Also, is this SHM?

Gold Member
I observe the equation is a non linear one and do not expect to find general solution easily.

phantomvommand
phantomvommand
I observe the equation is a non linear one and do not expect to find general solution easily.
Is it just the sum of a particular function and complementary function?

Gold Member
Say k=0 and ##\dot{x}_0>0## we get familiar relation of
$$\dot{x}=-\mu g t + \dot{x}_0$$
$$x=-\frac{1}{2}\mu g t^2 + \dot{x}_0 t+x_0$$
for 0<t<##\frac{\dot{x}_0}{\mu g}##, x= ##\frac{\dot{x}_0^2}{2\mu g}+x_0## for t beyond.

For k##\neq##0 similarly you can solve the equation until when ##\dot{x}=0##. Then for time beyond it change sign of friction term until next time of ##\dot{x}=0## and so on.

Last edited:
Mentor
Summary:: A spring has 1 end fixed to the wall, and the other end is connected to a block. Find the equation of motion of the block, given that it experiences only the spring force and a friction force = ##\mu mg##.

Is this SHM?
No. SHM does not have anything like the friction force.

phantomvommand