Problem $5.$ A uniform, $8\mathrm{.}0$ m long, $1500$ kg massive beam is hinged to a wall and supported by a thin cable attached $2\mathrm{.}0$ m from the free end of the beam $($see Fig. $3).$ The beam is supported at an angle of $30$ above the horizontal.

$($a$)$ Draw a free body diagram of the beam.

$($b$)$ Find the tension in the cable.

$($c$)$ How hard does the beam push inward on the wall?

Figure $3$: A beam hinged on a wall.

Problem $6.$ A beam resting on two pivots has a length of $L=6\mathrm{.}00m$ and mass $M=90\mathrm{.}0kg$ The pivot under the left end exerts a normal force vec$(F{)}_{N1}$ on the beam, and the second pivot placed a distance $l=4\mathrm{.}00m$ from the left end exerts a normal force vec$(F{)}_{N2}.$ A woman of mass $m=55\mathrm{.}0kg$ steps onto the left end of the beam and begins walking to

$2$

the right, as in Fig. $4.$ The goal is to find the woman's position when the beam begins to tip.

$($a$)$ Sketch a free$-$body diagram of the beam with the woman standing x$-$meters to the right of the first pivot.

$($b$)$ Where is the woman when the normal force vec$(F{)}_{N1}$ is the greatest?

$($c$)$ What is vec$(F{)}_{N1}$ when the beam is about to tip?

$($d$)$ Use the force equation of equilibrium to find the value of vec$(F{)}_{N2}$ when the beam is about to tip.

$($e$)$ Using the result of part c$)$ and the torque equilibrium equation, find the woman's position when the beam is about to tip.

$($f$)$ Check your answer by using a different axis of rotation.

Figure $4$: Woman tipping a beam.