I stumped with this math/physics question and I would appreciate if help can be given. Consider a horizontal spring attached to a block of mass m. Let the spring constant be k. On the surface of the spring there is friction whose coefficient is u. The spring is pulled a distance A from its equilibrium point. I derived a differential equation for this following system:
$$ F = ma$$ $$ -kx + umg = m \frac{d^2 x}{dt^2}$$ $$ \frac{d^2 x}{dt^2} + \frac{k}{m}x = ug, x(0) =A, x'(0)= 0$$
Since there is friction in the system, I would expect the spring to come to a halt after a certain time. However, the solution to this differential equation is
$$x(t) = (A- \frac{umg}{k})cos \sqrt{\frac{k}{m}} t +\frac{umg}{k}$$
The graph of this function, however, is purely sinusoidal and it does not tend to 0 as time approaches infinity. Is there something wrong with my differential equation? Thank you!
The problem is that your frictional force is not correct: The direction of the friction is opposite to the direction of the motion, i.e. $$F_{fr}=-\operatorname{sgn}(\dot{x})umg$$ (When $x=0$ is the equilibrium)
Your $F_0=umg$ force is just a constant force, it's like if you were to put it in constant gravitational field (that's why it only effects the equilibrium).