Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose one end of a rope…

Stretchable ropes are used to safely arrest
the fall of rock climbers. Suppose one end of a rope with unstretched length  is anchored to a cliff and a climber of mass m is attached to the
other end. When the climber is a height  above the anchor point, he slips and falls under the influence of
gravity for a distance 2, 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

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Stretchable ropes are used to safely arrest
the fall of rock climbers. Suppose one end of a rope with unstretched length  is anchored to a cliff and a climber of mass m is attached to the
other end. When the climber is a height  above the anchor point, he slips and falls under the influence of
gravity for a distance 2, 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

(b) Assuming m = 85 kg,  = 8.0 m and k = 850 N/m, determine x/ (the fractional stretch of the rope) and kx/m g (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

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