Harmonic motion with spring moving with constant speed

Feb 2019
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A spring whose constant is $k$ has its left end joined to a mass $m$ while its other end is always moving whith constant speed $u$. The coefficient of kinetic friction between the mass and the horizontal surface is μ and the coefficient of static friction is 2μ. The mass can carry out harmonic movements depending on the initial conditions. Analyze the two following:

1) First, consider that at $t=0$ the force of the spring is exactly equal to the kinetic friction force and the mass moves with speed $v_0 < u$. Find the time period $T_1$ and the amplitude $A$ of the oscillations of the spring.

2) Now consider that at $t=0$ the mass is at rest and the spring is not compressed. Graph the speed of the mass with respect to the floor from $t=0$ to $t=5T_2$, where $T_2$ is the time period of the oscillations de x(t), where $x(t)$ is the position of the mass measured from the initial point. Determine the average speed during its first 100 hundred time period of oscillations.

Sorry, if there are any grammatical mistake, I am not an english native.
 

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