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.

The crane can be adjusted for any angle

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

Solution:

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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\cdotm (clockwise)

 

Final Answers:

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

Maximum moment = 14715 N\cdotm (clockwise)

 

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 4, question 4-6.

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