Question

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 middleof the rope (Figure 1) . The rope will break if the tension in it exceeds 2.45×104N , and our hero's mass is 89.2kg .
(A) If the angle between the rope and the horizontal is theta = 11.7^circ, find the tension in the rope.
(B) What is the smallest value the angle theta can have if the rope is not to break?

Answer 1

1. The rope is weightless.
2. There is no tension in therope when it is stretched horizontally between the cliffs.

At midpoint the angle made by the rope = 180 - 2 x 11.7
=156.6 degrees.
The angle between the rope and the hero = 90+11.7 = 101.7

By Lami's theorem: t/sin101.7 = 82.3/sin156.6, where t - tension in rope

t = 82.3*sin101.7/sin156.6 = 202.92 kg

This is less than 2.45 x 104 so the hero (a bit foolish) is safe.

If the angle is theta, at break by Lami's theorem again:
82.3/sin(180- 2 theta) = (2.45 x 104)/sin(90+theta)

Simplifying and solving for theta: theta = degrees

Answer 2

let the angle by vertical is @
then
W/cos@ =T
T(max)=24500N
W=mg=89.2*9.8=874.16N
=>cos@=0.03568
=>@=87.95 degree