Static Friction Problem

[Lalo1985]

Hi, I'm having trouble solving this problem:

A woman attempts to push a box of books that has mass up a ramp inclined at an angle (alpha) above the horizontal. The coefficients of friction between the ramp and the box are (mu_k) and (mu_s). The force F applied by the woman is horizontal.

If (mu_s) is greater than some critical value, the woman cannot start the box moving up the ramp no matter how hard she pushes. Calculate this critical value of (mu_s).

I know that the maximum value of mu_s is f_mu/N. So, using F(y) = ma(y), I got: N - F*Gcos(alpha) - F*Gsin(alpha) = 0. Therefore, making N = F*Gcos(alpha) + F*Gsin(alpha). That's it. I don't know what to do next. Any ideas?

[maverick280857]

You have to resolve F in two directions: parallel to the incline and perpendicular to it. This will give you two equations, both involving F and one involving the normal reaction N, friction force f and acceleration a. Since the mass (of books) does not move up or down the incline, the acceleration a = 0. So this problem reduces to

\sum F_{x} = 0

\sum F_{y} = 0

where x and y are the directions parallel and perpendicular to the incline.

You should get the following equations as a result

F\cos\alpha - mg\sin\alpha - f_{s} = 0
N - mg\cos\alpha - F\sin\alpha = 0

Solve them to get the value of \mu_{s} using the fact that f_{s} = \mu_{s}N.

Hope that helps...

Cheers
Vivek

EDIT--The above equations are valid iff F is applied in the horizontal direction (i.e. in a direction parallel to the base of the incline).