Chapter 7, Problem 90GP

Stretchable ropes ate used to safely arrest the fall of rock climbers. Suppose one end of a rope with unstretched length ℓ is anchored to a cliff and a climber of mass m is attached to the other end. When the climber is a height ℓ above the anchor point, he slips and falls under the influence of gravity for a distance 2ℓ, after which the rope becomes taut and stretches a distance x as it stops the climber (see Fig. 7–33). Assume a stretchy rope behaves as a spring with spring constant k. (a) Applying the work-energy principle, show that

$x=\frac{mg}{k}\left[1+\sqrt{1+\frac{4k\mathcal{l}}{mg}}\right].$

(b) Assuming m = 85 kg, ℓ = 8.0 m and k = 850 N/m, determine x/ℓ (the fractional stretch of the rope) and kx/mg (the force that the rope exerts on the climber compared to his own weight) at the moment the climber’s fall has been stopped.

FIGURE 7–33

Problem 90.