Answer to Question #90483 in Mechanics | Relativity for brittany

Question

You have an adjustable, wooden incline and want to know how large of an angle would you need to raise the incline before several objects will slide down the incline. Determine the angles where the following object will begin to slide down the incline

Answer 1

Consider an object resting on an incline. Unless the body moves, three forces are acting on it: the force of gravity, normal force and force of friction.

For the considered moment according to Newton's second law for the equilibrium of forces we can write:

$$Ox:\space -mg\text{sin}\alpha=F_f,\\ Oy:\space -mg\text{cos}\alpha=N,$$

where the force of friction is

$$F_f=\mu_s N,$$

$\mu_s$ - a coefficient of static friction.

Re-write the two equations written for axes:

$$Ox:\space -mg\text{sin}\alpha=\mu_s N,\\ Oy:\space -mg\text{cos}\alpha=N,$$

divide the upper by the lower:

$$\frac{\text{sin}\alpha}{\text{cos}\alpha}=\mu_s,$$

or

$$\text{tan}\alpha=\mu_s,$$

and

$$\alpha=\text{atan}\mu_s.$$

This is the maximum angle at which the body can rest on an incline. Increase that angle and the body will begin to slide.