Question

Short answer

The tension force in each segment is \(2.409 \times {10^6}\;{\rm{N}}\).

Step-by-step solution

Given data

The angle is\(\theta = 10^\circ \).

The number of sides of the dome is\(N = 36\).

The required force on each segment is \(F = 4.2 \times {10^5}\;{\rm{N}}\).

Understanding Newton’s second law for horizontal and vertical directions

In this problem, while evaluating the tension that must exist in each segment, write Newton’s second law for the horizontal as well as vertical directions.

Free body diagram and calculation of forces in the horizontal direction

The following is the free-body diagram.

The relation to calculate the force can be written as:

\(\begin{array}{c}\sum {F_{\rm{x}}} = 0\\\left( {{F_{\rm{2}}} \times \cos \theta } \right) - \left( {{F_{\rm{1}}} \times \cos \theta } \right) = 0\\{F_{\rm{1}}} = {F_{\rm{2}}}\end{array}\)

Here, \({F_{\rm{1}}}\)and \({F_{\rm{2}}}\) are the tension forces on each segment of the dome.

Evaluation of tension force on each segment

The relation to calculate the tension force can be written as:

\(\begin{array}{c}\sum {F_{\rm{y}}} = 0\\\left( {{F_{\rm{1}}} \times \sin \theta } \right) + \left( {{F_{\rm{2}}} \times \sin \theta } \right) - {F_{\rm{B}}} = 0\end{array}\)

On plugging the values in the above relation, you get:

\(\begin{array}{c}\left( {{F_{\rm{1}}} \times \sin \theta } \right) + \left( {{F_{\rm{1}}} \times \sin \frac{\theta }{2}} \right) - {F_{\rm{B}}} = 0\\{F_1} = \frac{{{F_{\rm{B}}}}}{{2\sin \frac{\theta }{2}}}\\{F_1} = \left( {\frac{{4.2 \times {{10}^5}\;{\rm{N}}}}{{2\sin \frac{{10^\circ }}{2}}}} \right)\\{F_1} = 2.409 \times {10^6}\;{\rm{N}}\end{array}\)

Thus, \({F_1} = 2.409 \times {10^6}\;{\rm{N}}\) is the required force.