Question

Austin traverses a chasm by stringing a rope between a tree on one side of the chasm and a tree on the opposite side, 34 m away. Assume the rope can provide a tension force of up to 3.8 kN before breaking (k = 10^3). (a) If Bob’s mass is 5.0 kg, determine the angle θ that the string makes from horizontal when the rope is about to break. (b) Calculate the maximum distance x that the rope can sag without breaking.

Answer 1

Draw a diagram:

(a) As we see, the forces must be in equilibrium for rope not to break. Write this equilibrium:

$$mg=2T\text{ sin}\theta,\\\space\\ \theta=\text{arcsin}\frac{mg}{2T}=\text{arcsin}\frac{5\cdot9.8}{2\cdot3800}=0.37°.$$

(b) This distance is

$$d=\frac L2\text{ sin}\theta=\frac {34}2\text{ sin}0.37°=0.11\text{ m, or 11 cm}.$$