Question

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

Other

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?

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