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**[SOLVED] Challenging friction problem**

**1. Homework Statement**

A worker is shoving boxes up a rough plank inclined at angle [itex]\alpha[/itex] 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 distance [itex]x[/itex] along the plank: [itex]\mu = Ax[/itex], where [itex]A[/itex] is a positive constant and the bottom of the plank is at [itex]x = 0[/itex].For this plank, [itex]\mu_k = \mu_s = \mu[/itex]. The worker shoves the box up the plank so that is leaves the bottom of the plank moving at speed [itex]v_0[/itex]. Show that when the box first comes to rest, it will remain at rest if

[itex]v^2_0 \geq {3g\sin^2\alpha\over A\cos\alpha}[/itex]

**2. Homework Equations**

[tex] \vec{F}_{net} = \Sigma \vec{F} = 0[/tex]

[tex] \vec{F}_{net} = \Sigma \vec{F} = m \vec{a}[/tex]

[tex]f_s <= \mu_s n[/tex]

[tex]f_k = \mu_k n[/tex]

[tex]w = m g[/tex]

**3. The Attempt at a Solution**

initial speed = [itex]v_0[/itex]

final speed = [itex]v[/itex]

taking [itex]xy[/itex] axes along and perpendicular to the plank:

[itex]n = mg\cos\alpha[/itex]

[tex]

\begin{align*}

v - mg\sin\alpha - f = -ma\\

v - mg\sin\alpha - \mu mg \cos\alpha = -ma\\

mg\sin\alpha + \mu mg \cos\alpha - v = ma\\

mg\sin\alpha + Axmg\cos\alpha - v = ma\\

\end{align*}

[/tex]

I get stuck here now, I don't even think I'm heading in th right direction.

I've tried the problem using the work-energy theorem also but I can't get anything to cancel down to the equation needed.

Any help gratefully received

thanks, b.