Question

Determine the angle to the nearest degree at which a desk can be tilted before a paperback book on the desk begins to slide. Use the equation mg sin A = umg cos A and assume u = 1.11.

Answer 1

The point of this exercise is to find the angle at which the net forces in both directions are equal to 0. That's the point where the book won't move...and any angle greater than that will cause it to move.

While at rest:\[\large F_{netX}=mgsin(\theta)-\mu F_N=0N\]\[\large F_{netY}=F_N-mgcos(\theta)=0N\]
Rearranging the above you can see what the normal force equals:\[\large F_N=mgcos(\theta)\]
Plug that back into the first equation to get:\[\large F_{netX}=mgsin(\theta)-\mu mgcos(\theta)=0N\]\[\large F_{netX}=mg(sin(\theta)-\mu cos(\theta)=0N\]\[\large sin(\theta)-\mu cos(\theta)=0\]\[\large sin(\theta)=\mu cos(\theta)\]\[\large \frac{sin(\theta)}{cos(\theta)}=\mu\]\[\large tan(\theta)=\mu\]\[\large arctan(\mu)=\theta\]Plugging the coefficient of static friction you were given into the last equation above and you get:\[\large arctan(1.11)=47.98^\circ\]Therefore, any angle > 47.98 degrees will result in a force that overcomes the force of static friction causing the block to begin sliding.

Answer 2

Thank you very much! :D

Answer 3

yw :)