A box of mass m is at rest on a horizontal floor. The coefficients of static and kinetic friction between the box and the floor are \mu_{0} and \mu (less than \mu_{0} ), respectively. One end of a spring with spring constant k is attached to the right side of the box, and the spring is initially held at its relaxed length. The other end of the spring is pulled horizontally to the right with constant velocity v_{0}. As a result, the box will move in fits and starts. Assume the box does not tip over.
a. Calculate the distance s that the spring is stretched beyond its rest length when the box is just about to start moving.
b. Let the box start at x=0, and let t=0 be the time the box first starts moving. Find the acceleration of the box in terms of x, t, v_{0}, s, and the other parameters, while the box is moving.
The position of the box as a function of time t as defined in part (b) is
where \omega=\sqrt{k / m} and r=\mu / \mu_{0}. This expression applies as long as the box is still moving, and you can use it in the parts below. Express all your answers in terms of v_{0}, \omega, s, and r.
c. Find the time t_{0} when the box stops for the first time.
d. For what values of r will the spring always be at least as long as its rest length?
e. After the box stops, how long will it stay at rest before starting to move again?