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Magazine Chapter 9: Q10PE (page 316)
A \({\rm{17}}{\rm{.0}}\;{\rm{m}}\)high and \({\rm{11}}{\rm{.0}}\;{\rm{m}}\)long wall under construction and its bracing are shown in Figure \({\rm{9}}{\rm{.32}}\). The wall is in stable equilibrium without the bracing but can pivot at its base. Calculate the force exerted by each of the 10 braces if a strong wind exerts a horizontal force of \({\rm{650}}\;{\rm{N}}\)on each square meter of the wall. Assume that the net force from the wind acts at a height halfway up the wall and that all braces exert equal forces parallel to their lengths. Neglect the thickness of the wall.
Short Answer
The force exerted by each of the braces is \({\rm{21193}}\;{\rm{N}}\).
Step by step solution
Force
A force can change the rest or motion of a body in a general direction.
Diagram
The length of the wall is \({{\rm{l}}_{\rm{w}}}{\rm{ = 11}}\;{\rm{m}}\).
The height of the wall is \({\rm{h = 17}}\;{\rm{m}}\).
The force per square meter is \({\rm{f = 650}}\;{\rm{N}}\)
The free-body Diagram
Calculation of the force
The torque by the wind force is,
\(\begin{array}{c}{\tau _{wind}} = f \times A \times {r_{wind}} \times \sin 90^\circ \\ = 650 \times 11 \times 17 \times 8.5 \times 1\\ = 1033175\;N\end{array}\)
Torque due to braces is,
\(\begin{array}{c}{{\rm{\tau }}_{\rm{b}}}{\rm{ = The}}\;{\rm{number}}\;{\rm{of}}\;{\rm{braces \times F \times }}{{\rm{r}}_{{\rm{brace}}}}{\rm{ \times sin35^\circ }}\\{\rm{ = 10 \times F \times 8}}{\rm{.5 \times sin35^\circ }}\\{\rm{ = 48}}{\rm{.75F}}\end{array}\)
For equilibrium, this two torque will be the same.
\(\begin{array}{c}{\rm{48}}{\rm{.75F = 1033175}}\\{\rm{F = 21193}}\;{\rm{N}}\end{array}\)
Hence, the force is \({\rm{21193}}\;{\rm{N}}\).
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