The Leaning LadderA uniform ladder of length { rests against a smooth, vertical wall (figure (a)). The mass of the ladder is m, and the coefficient ofstatic friction between the ladder and the ground is H = 0.46. Find the minimum angle 0min at which the ladder does not slip.(a) A uniform ladder at rest,leaning against a smooth wall.The ground is rough.(b) The forces on the ladder.mgSOLUTIONConceptualize Think about any ladders you have climbed. Do you want a large friction force between the bottom of the ladder andthe surface or a small one? If the friction force is zero, will the ladder stay up? Simulate a ladder with a ruler leaning against avertical surface. Does the ruler slip at some angles and stay up at others?Categorize We do not wish the ladder to slip, so we model it as a rigid object in -Select--Analyze A diagram showing all the external forces acting on the ladder is illustrated in figure (b). The force exerted by the groundon the ladder is the vector sum of a normal force n and the force of static friction f..The wall exerts a normal force P on top of theladder, but there is no friction force here because the wall is smooth. So the net force on the top of the ladder is perpendicular tothe wall and of magnitude P.Apply the first condition for equilibrium to the ladder in both the x- and the y-directions (For Equations (1) - (5), use the followingas necessary: m, g, f, (lowercase subscript), and l.):(1) E, - - P = 0(2) F, - n -Solve Equation (1) for P:(3) P-Solve Equation (2) for n:(4) n-When the ladder is on the verge of slipping, the force of static friction must have its maximum value, which is given byfs, max = Hn. Combine this equation with Equations (3) and (4):(5) Pmax = fs, max = Hgn = HsApply the second condition for equilibrium to the ladder, evaluating torques about an axis perpendicular to the page through O (Usethe following as necessary: 0, P, and f.):- mg cos(0) = 02Solve for tan(@) (Use the following as necessary: P, m, g, and e.):sin(0)= tan(0) =mg+ 0 = tan2Pcos(e)Under the conditions that the ladder is just ready to slip, e becomes eminand Pmax is given by Equation (5). Substitute:Omin = tan-1(-ng = tan=maxFinalize Notice that the angle depends only on the -Select--not on the mass or length of the ladder.EXERCISEIf the coefficient of static friction is 0.60, and the same ladder as in the example makes a 53° angle with respect to the horizontal,how far (in m) along the length of the ladder can a 58.0 kg painter climb before the ladder begins to slip? The ladder is 7.5 m inlength and has a mass of 21 kg.HintmNeed Help?Read It

A uniform ladder of length f rests against a smooth, vertical wall (figure (a)). The mass of the ladder is m, and the coefficient of static friction between the ladder and the ground is = 0.46. Find the minimum angle emin at which the ladder does not slip.

A uniform ladder of length f rests against a smooth, vertical wall (figure (a)). The mass of the ladder is m, and the coefficient of static friction between the ladder and the ground is = 0.46. Find the minimum angle emin at which the ladder does not slip.

Mechanics of Materials (MindTap Course List)

9th Edition

ISBN:9781337093347

Author:Barry J. Goodno, James M. Gere

Publisher:Barry J. Goodno, James M. Gere

Chapter2: Axially Loaded Members

Section: Chapter Questions

Problem 2.8.6P

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