A mass m on a horizontal surface with mild kinetic friction mu_L is attached to

Question

A mass m on a horizontal surface with mild kinetic friction mu_L is attached to an ideal spring with spring constant k. After the spring is compressed by a distance d to the left of its equilibrium position and released (as shown above), the mass starts to slide look at the right. Express your final answers ONLY in terms of k, m, d, mu, the variable x, and any necessary physical and mathematical constants. Find an expression for the acceleration of the mass as a function of its position x (namely, a(x)) as the mass slides back to its equilibrium position (between x = -d and x = 0). Show your work. For this question, ignore any state friction that may be present when the mass starts from rest. Assume that kinetic friction is weak enough that the mass actually does accelerate to the right after being released. Use your answer for part (a) to find the position x (between -d and 0) where the mass's acceleration momentarily equals zero. Show your work. What is the minimum coefficient of static friction need so that the mass would remain at rest when it is released at x = -d? Show your work.

Explanation / Answer

a)   Net force on block = Force provided by spring -   kinetic frictional force

=>     m*a = k*x   - kmg

=>     a(x) = k*x/m - kg

b)    For zero acceleration

    a(x)   = 0

=>   k*x/m - kg = 0

=>   k*x/m = kg

   =>   x = - mkg / k                -------------------->   position where mass acceleration is zero

c)     For mass to remain at rest

     smg = k*d

   =>   s = k*d/mg             --------------------> minimum coefficient of static friction

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