Accelerating Incline - Conditions to prevent slipping

[Pi-Bond]

Homework Statement

(Kleppner & Kolenkow - Introduction to Mechanics - 2.17)
The first attached image shows a box on an incline which is accelerated at a rate a meters per second squared. $\mu$ is the coefficient of friction between the box and incline surface. The questions are in the image.

Homework Equations

Newton's Laws

The Attempt at a Solution

I can understand that there will be a range for the accelerations which will leave the box static on the incline. I'm not sure how the math is to be applied to get the maximum acceleration. I got the answer for minimum by resolving forces along the surface of the incline, and equating them. The answer I got was:

$a=g \frac{sin(\theta)-\mu cos(\theta)}{cos(\theta)+ \mu sin(\theta)}$

This gives the correct answer for the case given in the clues. I'm not sure how to proceed to find the maximum acceleration. Any hints?

 

[Doc Al]

Hint: Consider the direction of the friction force.

 

[Pi-Bond]

Of course! In the maximum case, the block is on the verge of drifting up the incline, so the friction force must be directing down the incline! So the maximum acceleration is given by-

$a=g \frac{sin(\theta)+\mu cos(\theta)}{cos(\theta)-\mu sin(\theta)}$

Thanks a lot!

 

[aLearner]

I got the same answer, but solved for μ. So

μ = $\frac{gsinθ - acosθ }{asinθ + gcosθ }$

but the correct answer is μ = tanθ
help?