Coefficient of static friction?

This question was just on my test and i really want to know how it is done...


A wooden block is sitting on an surface, as the surface begins to tilt, the wooden block does not move until the surface reaches an incline of 26.5 degrees, what is the coefficient of static friction?


a) .4


b) .5


c) .6


d) not enough information is given


e) none of the above





I'm more interested in an explanation then the actual answer.

Coefficient of static friction?
If I remember this correctly..





Ff = Cf*N





Once you have the angle, then





Ff = Fg





Fg = ma*sin(26.5)


Ff = Cf*ma*cos(26.5)





ma*sin(26.5) = Cf*ma*cos(26.5)


tan(26.5) = Cf = 0.50
Reply:The force acting from the surface on the block is not always perpendicular to the surface. This force can tilt with respect to the perpendicual, but only up to certain critical angle θc. Tangent of this critical angle is called coefficient of friction:


μ = tan(θc).





Force of reaction acting on the block at rest must be opposite to the force of gravity mg acting on the block. Once you tilt the surface beyond the critical angle θc, the force of reaction cannot tilt that much, and the block bigins to slide.





Answer:


μ = tan(θc) = tan(26.5 deg) = 0.5
Reply:As usual Alex is right on point. But perhaps you'd like to see where the tangent factor came from.





Suppose a friction force F = kN = k mg cos(theta); where k is the friction coefficient, m is the mass (your block), g ~ 9.81 m/sec^2 in kms units, and theta is the incline of the tilting ramp.





Now suppose gravity pulling against F to start the block sliding is w = W sin(theta) = mg sin(theta); where W = mg is the weight of the block m.





Then the net forces acting on the block along the surface of the ramp are f = ma = w - F. Therefore, the block will start to accelerate (start to slide) when a %26gt; 0 the acceleration along the surface of the ramp. And for this to be, we have w - F %26gt; 0 or w = mg sin(theta) %26gt; k mg cos(theta) = F [See where this is going?]





Solve for k, the critical static coefficient, so we have sin(theta)/cos(theta) %26gt; k or, ta da, tan(theta) %26gt; k; where theta = 26.5 deg in your problem. And there you have it, the why of Alex's answer.