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Friction

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Say you have two beams at different height and a plank with length L and a centre of gravity that is fL where f is a fraction. Both the beams have variable coefficients of friction, u1 and u2. How can i figure out the angle of the plank where it starts to slip? Heres what I have so far: The mass needs to be distributed across the two beams. So beam one has (1 - f)*mass resting on it and beam two has f*mass resting on it. The maximum static friction forces are given by u1*mgcos(theta) and u2*mgcos(theta) for each of the beams, where m depends on the beam (as above). The force pulling in the other direction is mgsin(theta). So the the system slips when mgsin(theta) > u1*mgcos(theta) + u2*cos(theta) So that means I have to differentiate somehow to get the maximum value for theta. Am I on the right track or missing something completely?

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You shouldnt need to differentiate. Rather, think about the point exactly when the beam will start to slip. The force pulling the beam down will be equal to the maximum amount of friction. So you'll have

mgsin(theta) = u1*(1 - f)*mgcos(theta) + u2*fmg*cos(theta)

I think you'll be able to do the algebra from there

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