Engineering Mechanics: Statics and Dynamics, 14th Edition
Authors: Russell C. Hibbeler
ISBN-13: 978-0133915426
See our solution for Question 11P from Chapter 4 from Hibbeler's Engineering Mechanics.
Problem 11P
Step-by-Step Solution
We have given that the magnitude of the force acting at the end of the boom is $P = 6\;{\rm{kN}}$.
We are asked to calculate the distance $x$ and the moment about point O for the maximum moment condition.
Step 2
For the maximum moment, the cable AB must be perpendicular to the boom AO.
A labeled diagram for the maximum moment condition is shown below.
Calculate the distance ${x_1}$ from the geometry of the diagram:
\[\begin{array}{c} \cos 30^\circ = \frac{{8\;{\rm{m}}}}{{{x_1}}}\\ {x_1} = \frac{{8\;{\rm{m}}}}{{\cos 30^\circ }}\\ {x_1} = 9.24\;{\rm{m}} \end{array}\]Calculate the distance ${x_2}$ from the geometry of the diagram:
\[\begin{array}{c} \tan 30^\circ = \frac{{{x_2}}}{{1\;{\rm{m}}}}\\ {x_2} = 1\;{\rm{m}} \times \tan 30^\circ \\ {x_2} = 0.58\;{\rm{m}} \end{array}\]The distance $x$ can be calculated as:
\[x = {x_1} + {x_2}\]Substitute the value of ${x_1}$ and ${x_2}$ in the above equation:
\[\begin{array}{c} x = 9.24\;{\rm{m}} + 0.58\;{\rm{m}}\\ = 9.82\;{\rm{m}} \end{array}\]Step 3
Consider the moment in the counter-clockwise direction as positive.
The equation of moment about point O can be given as:
\[{M_O} = P \times 8\;{\rm{m}}\]Substitute the value of $P$ in the above equation:
\[\begin{array}{c} {M_O} = 6\;{\rm{kN}} \times 8\;{\rm{m}}\\ = 48.0\;{\rm{kN}} \cdot {\rm{m}} \end{array}\]