Block sliding down wedge max value problem

[bodensee9]

Hello:


I have an inclined triangular wedge of angle Ø (meaning, this wedge makes angle Ø with the horizontal) and a block is lying on the wedge. the coefficient of friction between the wedge and block is u.

(a) determine the max value of Ø for the block to remain motionless on the wedge when the wedge is fixed in position.
Let m = mass of block, then
This part is just umgcosØ = mgsinØ. So then tanØ = u, and u = arctanØ

(b) the wedge is given horizontal acceleration a. Assuming that tanØ < u, find the maximum and minimum acceleration so that the block doesn't slide. I think for the minimum, I will just set friction along the wedge = the component of acceleration along the wedge + the component of gravity on the block along the wedge. But I'm not sure what to do about finding the maximum acceleration? I think not sliding means that they are both accelerating horizontally at a.

Thanks.

 

[LowlyPion]

For the maximum, they mean not accelerating up the incline.

Remember that friction is bi-directional. So for the maximum, you need to overcome via acceleration, both the downward component of gravity along the incline as well as the friction resisting any motion.

 

[RoyalCat]

A good way I've found to approach these problems is to think in terms of how much force the static friction can enact on the block.
Remember that f_{s max} = µ_s * N = µ_s*mg*cosΘ

Also, not sliding would mean that the block is at rest relative to the wedge. If you know how to work with fictitious forces, I suggest you draw a free body diagram of the block, from the accelerated frame of the block, with fictitious forces, and say that the block is at rest and see where that gets you.

A good point to make here (And please, someone, correct me if I'm wrong) is that you should also take note of two cases. One where without friction the block would slide down the slope, and one where without friction the block would slide UP the slope. Remember that f_s is a vector, and its direction is opposite that of the expected velocity.

 

[bodensee9]

oh right, that helps. i think the difference is that friction points in opposite directions. thanks.

 

[shadowplay]

Hi,

Even though I know how to 'solve' this problem mathematically, something about it still bothers me, physically speaking. So having calculated the minimum and maximum accelerations (lets call them a_min and a_max), suppose we give the block an acceleration a strictly between these two values. i know the block won't move relative to the wedge. But how can we know the direction of (static) friction (upwards or downwards)?

Thanks