Serway & Jewett — Static Equilibrium and Elasticity: Problem 8

A uniform beam of length \( L \) and mass \( m \) shown in Figure P12.8 is inclined at an angle \( \theta \) to the horizontal. Its upper end is connected to a wall by a rope, and its lower end rests on a rough, horizontal surface. The coefficient of static friction between the beam and surface is \( \mu_s \). Assume the angle \( \theta \) is such that the static friction force is at its maximum value. (a) Draw a force diagram for the beam. (b) Using the condition of rotational equilibrium, find an expression for the tension \( T \) in the rope in terms of \( m \), \( g \), and \( \theta \). (c) Using the condition of translational equilibrium, find a second expression for \( T \) in terms of \( \mu_s \), \( m \), and \( g \). (d) Using the results from parts (a) through (c), obtain an expression for \( \mu_s \) involving only the angle \( \theta \). (e) What happens if the ladder is lifted upward and its base is placed back on the ground slightly to the left of its position in Figure P12.8? Explain.