A block lies on a wedge with the slope angle `alpha`. The coefficient of static friction between the bar and the wedge is `mu lt tan alpha`. What should be the acceleration of the wedge to prevent the bar from sliding down?

Answer

`g tan (alpha-varphi) le a le g tan (alpha+varphi)`, where `varphi=` arctan `mu`

Explanation

Since the block does not slide on the wedge, the direction of the force of kinetic friction is not known. Evidently, if the acceleration of the wedge is small, the block will slide downwards and the friction force will be directed as shown in Fig.. When the acceleration of the block is large, the wedge slides upwards and the direction of the block of friction changes sign (see Fig.).
Write down the equations of motion along the coordinate axes for both cases:
y-axis : `Q cos alpha + T sin alpha - mg = 0, Q cos alpha-T sin alpha - mg = 0`
x-axis : `Q sin alpha - T cos apha = ma_(1), Q sin alpha + T cos alpha = ma_(2)`
Noting that `T= muQ`, we obtain after some transformations:
`a_(1)= "g" (tan alpha-mu)/(1+mu tan alpha), a_(2)= "g"(tan alpha +mu)/(1-mu tan alpha)`
Writing `mu=tan varphi`, we obtain `g tan (alpha-varphi) le a le g tan (alpha +varphi)`.