L. M Figure P12.24

A uniform beam resting on two pivots has a length L = 6.00 m and mass M = 90.0 kg. The pivot under the left end exerts a normal force η₁ on the beam, and the second pivot located a distance ℓ = 4.00 m from the left end exerts a normal force η₂. A woman of mass m = 55.0 kg steps onto the left end of the beam and begins walking to the right as shown. 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 η₁ is the greatest? (d) What is η₁ when the beam is about to tip? (e) Use as shown to find the value of η2 when the beam is about to tip.
(f) Using the result of part (d) and as shown, 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.