Question

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

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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)

2021-02-21
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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