A classical mechanics problem by Vishal Gupta

On a winter day in Maine, a warehouse worker is shoving boxes up a rough plank inclined at an angle \(\alpha\) above the horizontal. The plank is partially covered with ice, with more ice near the bottom of the plank than near the top, so that the coefficient of friction increases with the distance \(x\) along the plank: \(\mu = Ax\), where \(A\) is a positive constant and the bottom of the plank is at \(x=0 \). (For this plank the coefficients of kinetic and static friction are equal: \( \mu _k = \mu_ s = \mu\).) The worker shoves a box up the plank so that it leaves the bottom of the plank moving at speed \(v_0\). Show that when the box first comes to rest, it will remain at rest if \(\text{__________} \).

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