What is the force needed to make the wedge move?

[stinlin]

Yeah - so this WAS on our homework, but our professor was even stumped at first...Eventually he figured it out, but he told us not to worry about it.

But of course, I'm SUPER curious! Can someone help me figure this out?

http://img120.imageshack.us/img120/4610/staticskg2.gif

It's a 10 degree wedge with an 80 kg cylinder resting on it. The coefficient of static friction between all surfaces of contact is 0.25. You need to determine the force P so that the motion of the wedge is impending.


Please help! I'm super curious as to how the HECK you're supposed to figure this out. =) Thanks!

 

[jdlawlis]

solution

In this problem you have three unknown normal forces, three unknown friction forces and one unknown applied force P. You will get two equations for the x and y components of the net force on the ball, two equations for the x and y components of the net force on the wedge, one equation for the net torque on the ball, and one equation for the net torque on the wedge. The net torques and net forces are all equal to zero. Your seventh equation will involve setting the friction force between the wedge and the ramp equal to mu_s*F_n. Seven variables -- Yikes! Sounds like matrices will come in handy.

 

[andrevdh]

By taking moments about the point where the cylinder touches the wall it is possible to solve for the friction that the cylinder experiences from the wedge, $$f$$ (same as the friction on the wedge from the cylinder), and therefore for the normal force, $$N$$, on the wedge from the cylinder

$$\Gamma _w = \Gamma _f + \Gamma _N$$

which comes to

$$w = f(1 + \cos(\theta)) + \frac{f\cos(\theta)}{\mu}$$

(the radius of the cylinder cancels out)