Problem $2.$ Cranes.

Relevant learning goals:

$-$ Calculate the torque due to a force about a reference point or axis.

Subskills: $1,2$

Cranes exhibit many of the physics concepts of this course: Newton's laws for the forces involved, pulleys to amplify the force on the load, rotational motion of the pulleys $($and sometimes flywheels to store energy$),$ gravitational potential energy, and, relevant for this problem, the torques involved in keeping it all balanced.

$($a$)$ To prevent the crane from falling over, cranes often have a counterweight on the opposite side of the load. What is the net torque at $\backslash ($ P$\_\{1\}\backslash )$ and at $\backslash ($ P$\_\{2\}\backslash )?$ Assume the structure $\backslash (\{\}^\{1\}\backslash )$ of the crane is much less massive than the loads so that you can ignore the weight of the horizontal booms.

$($b$)$ Now consider a crane that extends outward with a support cable instead of a counterweight. The cable connects to the boom $($the arm that supports the load$)$ a distance $\backslash ($ x $\backslash )$ from the tower and connects to the tower head at a distance $\backslash ($ y $\backslash )$ above the boom. What is the tension in the support cable?

$\backslash (\{\}^\{1\}\backslash )$ Real cranes, of course, are quite massive to ensure rigidity and prevent bending. If you take more advanced engineering courses, you will learn how to calculate the second moment of inertia $($also known as the area moment of inertia$)$ to analyze the resistance to bending.

$($c$)$ In order to minimize the tension in the cable, you are free to set $\backslash ($ x $\backslash )$ to be anything between $0$ and $\backslash ($ L $\backslash )$ and $\backslash ($ y $\backslash )$ to be anything between $0$ and $\backslash ($ H $\backslash ).$ What is the minimum tension in the cable?