Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose o
ID: 1316516 • Letter: S
Question
Stretchable ropes are 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 the figure(Figure 1) ). Assume a stretchy rope behaves as a spring with spring constant k.
art B
Assuming m = 85kg , l = 9.5m and k = 890N/m , determine x/l (the fractional stretch of the rope) at the moment the climber's fall has been stopped.
Express your answer using two significant figures.
Part C
Assuming m = 85kg , l = 9.5m and k = 890N/m , determine 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.
Explanation / Answer
B)
Apply enrgy consrvation
Initial graviational PE = final Elastic PE
m*g*(2*L+x) = 0.5*k*x^2
85*9.8*(2*9.5+x) = 0.5*890*x^2
15827 + 833*x = 445*x^2
x^2 - 1.872*x - 35.57 = 0
soving the above equation we get,
x = 6.97 m
x/L = 6.97/9.5
= 0.734 <<<<<<------Answer
C) k*x/m*g = 890*6.97/(85*9.8)
= 7.45 <<<<<<------Answer