An adventurous archaeologist crosses between two rock cliffs by slowly going hand-over-hand along a rope stretched between the cliffs.

asked 2021-02-20

An adventurous archaeologist crosses between two rock cliffs by slowly going hand-over-hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope (Fig. 1) The rope will break if the tension in it exceeds \(4.50\cdot10^4 \)N and our hero's mass is 86.5 kg.

 Figure 1 a) If the angle \(\theta\) is \(13.0^\circ\), find the tension iin the rope.

b) What is the smallest value the angle \(\theta\) can have if the rope is not to break?


Answers (1)

Each half of the rope exerts a vertical force of \(\displaystyle{T}\cdot{\sin{{\left(\theta\right)}}}\)
or \(\displaystyle{2}\cdot{T}\cdot{\sin{{\left(\theta\right)}}}={M}\cdot{g}\)
So \(\displaystyle{T}={\left({\left({86.5}\ {k}{g}\cdot{9.8}\ {\frac{{{m}}}{{{s}^{{2}}}}}\ \right)}{2}\cdot{\sin{{13}}}\right)}={1884}\ {N}\)
For the minimum angle \(\displaystyle{\sin{{\left(\theta\right)}}}={M}\cdot{\frac{{{g}}}{{{2}}}}\cdot{T}\)
\(\displaystyle={\frac{{{847.7}\ {N}}}{{{9}}}}\cdot{10}^{{{4}}}\ {N}\) So \(\displaystyle\theta\) minimum =.54deg
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