Serway & Jewett — Static Equilibrium and Elasticity: Problem 24

A uniform beam resting on two pivots has a length \(L = 6.00\text{ m}\) and mass \(M = 90.0\text{ kg}\). The pivot under the left end exerts a normal force \(n_1\) on the beam, and the second pivot located a distance \(\ell = 4.00\text{ m}\) from the left end exerts a normal force \(n_2\). A woman of mass \(m = 55.0\text{ kg}\) steps onto the left end of the beam and begins walking to the right as in Figure P12.24. The goal is to find the woman’s position when the beam begins to tip. (a) What is the appropriate analysis model for the beam before it begins to tip? (b) Sketch a force diagram for the beam, labeling the gravitational and normal forces acting on the beam and placing the woman a distance \(x\) to the right of the first pivot, which is the origin. (c) Where is the woman when the normal force \(n_1\) is the greatest? (d) What is \(n_1\) when the beam is about to tip? (e) Use Equation 12.1 to find the value of \(n_2\) when the beam is about to tip. (f) Using the result of part (d) and Equation 12.2, with torques computed around the second pivot, find the woman’s position \(x\) when the beam is about to tip. (g) Check the answer to part (e) by computing torques around the first pivot point.