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50. Problem 13.40P (HRW) A uniform plank, with length L=20 ft and weight W=100
lb, rests on the ground and against a frictionless roller at the top of a
wall of height h = 10 ft. The plank remains in equilibrium for any value
of
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but
slips if
. We
have to find the coefficient of static friction between the plank and the
ground.
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Solution: Click For PDF Version We will consider the free-body diagram of the plank. As the roller is
frictionless, it exerts a normal force N on the plank at the point of
contact. At the contact point of the plank with the ground there will be a
horizontal force
In the problem it has been given that when
From these equations we can solve for
In this relation on substituting
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due to
friction, which resists the sliding back of the plank, and normal force
acting in the vertical direction. 
,
the maximum force
, where
is the
coefficient of static friction between the plank and the ground. Conditions of
equilibrium are that the vector sum of the external forces has to be zero and
torque due to external forces about any point has to be zero. We will compute
the torque about the point where the plank is in contact with the ground. We get
the following algebraic relations because of these conditions. 
. We get
