# The crane can be adjusted for any angle

The crane can be adjusted for any angle 00 ≤ θ ≤ 900 any extension 0 ≤ x ≤ 5 m. For a suspended mass of 120 kg, determine the moment developed at A as a function of x and θ. What values of both x and θ develop the maximum possible moment at A? Compute this moment. Neglect the size of the pulley at B.

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

#### Solution:

Let us write an equation to calculate the moment using $\theta$ and $x$ as variables around point A.

$\circlearrowright M_A=((9-1.5)+x)(\cos\theta)(120)(9.81)$

Moment is the force multiplied by the perpendicular distance from the point we are referencing. Thus, in this question, the distance changes based upon the values of $x$. In addition, the angle also determines the length, as a small angle would allow for a longer perpendicular distance between the force that pulls on the crane and point A.

Simplifying our equation gives us:

$\circlearrowright M_A=(7.5+x)1177.2\cos\theta$

The maximum moment will develop if $\theta$ = $0^0$ and $x=5$ m. Substituting these values gives us:

$\circlearrowright M_A=(7.5+x)1177.2\cos\theta$

$M_A=(7.5+5)1177.2\cos0^0$

$M_A=14715$ N$\cdot$m (clockwise)

$\circlearrowright M_A=(7.5+x)1177.2\cos\theta$
Maximum moment = $14715$ N$\cdot$m (clockwise)