- #1

Silicon-Based

- 51

- 1

## Homework Statement

A block of mass [itex]m[/itex] is placed on a rough wedge inclined at an angle [itex]α[/itex] to the horizontal, a distance [itex]d[/itex] up the slope from the bottom of the wedge. The coefficient of kinetic friction between the block and wedge is given by [itex]µ_0x/d[/itex], where [itex]x[/itex] is the distance down the slope from the starting point. Calculate the maximum value of [itex]µ_0[/itex] which will allow the block to reach the bottom of the wedge.

**2. The attempt at a solution**

After drawing a diagram and resolving the forces I found the acceleration of the block in terms of [itex]x[/itex]:

$$a(x)=g(\sin(\theta)-x\frac{µ_0}{d}\cos(\theta))$$

I don't know how to proceed further, given that the acceleration is defined over [itex]x[/itex] rather than [itex]t[/itex], which prevents me from simply integrating this expression. I suspect the chain rule could be useful ([itex]a=\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}[/itex]), but here again, I don't know how [itex]x[/itex] depends on [itex]t[/itex].