Thread: Inclined plane problem View Single Post
P: 4,006
 Quote by tricky_tick can you please explain how you arrived at that conclusion?
You have four forces. Suppose the x-acis is along the ramp and the force F for pulling the object is aligned along this x-axis. Y-axis perpendicular to x-axis.

gravity
$$-mg\sin(\theta)*e_{x} - mg\cos(\theta)*e_{y}$$

normal force
$$N * e_{y}$$

friction
$$-{\mu}N*e_{x}$$

pull-force
$$F*e_{x}$$

The sum of y-componets yield :

$$N = mg\cos(\theta)$$

The sum of x-componets yield :

$$-{\mu}N -mg\sin(\theta) + F = 0$$

So we have that $$F = {\mu}mg\cos(\theta) + mg\sin(\theta)$$

or $$F = mg(\mu\cos(\theta) + \sin(\theta))$$

let's take the derivative with respect to the angle theta and set this equal to 0. Thus we get :

$$0 = mg(-{\mu}\sin(\theta) + \cos(\theta))$$
or
$$0 =-{\mu}\sin(\theta) + \cos(\theta)$$

or
$$\mu = \frac {\cos(\theta)}{\sin(\theta)} = \cot(\theta)$$
Thus $$\tan(\theta) = \frac {1}{\mu}$$

marlon.