10 STATICS
10.5 Ladders and walls wall
ladder
S workman Mg
l
x
R ground f
θ
Figure 93: A ladder leaning against a vertical wall.
a distance x along the ladder, measured from the bottom. See Fig. 93. Suppose that the wall is completely frictionless, but that the ground possesses a coefficient of static friction µ. How far up the ladder can the workman climb before it slips along the ground? Is it possible for the workman to climb to the top of the ladder without any slippage occurring? There are four forces acting on the ladder: the weight, M g, of the workman; the reaction, S, at the wall; the reaction, R, at the ground; and the frictional force, f, due to the ground. The weight acts at the position of the workman, and is directed vertically downwards. The reaction, S, acts at the top of the ladder, and is directed horizontally (i.e., normal to the surface of the wall). The reaction, R, acts at the bottom of the ladder, and is directed vertically upwards (i.e., normal to the ground). Finally, the frictional force, f, also acts at the bottom of the ladder, and is directed horizontally. Resolving horizontally, and setting the net horizontal force acting on the ladder to zero, we obtain S − f = 0. (10.28) Resolving vertically, and setting the net vertically force acting on the ladder to zero, we obtain R − M g = 0. (10.29) Evaluating the torque acting about the point where the ladder touches the ground, we note that only the forces M g and S contribute. The lever arm associated with the force M g is x cos θ. The lever arm associated with the force S is l sin θ. Fur227