Stretchable ropes are used to safely arrest the fall of rock c...

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 force of gravity for a distance 2ℓ , after which the rope becomes taut and stretches a distance x as it stops the climber (see Fig. 7–37). <IMAGE>

Assume a stretchy rope behaves as a spring with spring constant k.

(a) Applying the work-energy principle, show that

x = mg/k [ 1 + √1 + (4kℓ/mg)]

Assume a stretchy rope behaves as a spring with spring constant k.

(b) Assuming m = 85kg , 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.

Accepted Answer

Hey everyone in this problem, a thrill seeker plans to perform a safe jump off a 30 m high platform using an elastic rope. The individual has a mass of 70 kg and the jump involves a free fall of 60 m in total before the elastic rope which has an uns stretched length of 30 m begins to stretch to arrest the fall. The rope with a spring constant of 50 newtons per meter is carefully selected to ensure the jumper is safely brought to a stop without touching the ground. We're asked to calculate the maximum extension of the rope during the jump to ensure the safety of the thrill seeker. We're given a diagram of the situation we're gonna come back to in just a minute and we have four answer choices all in meters. Option A 31.2 option B 44.8 option C 56.6 and option D 62.4. If we take a look at this diagram, what we have is this thrill seeker on a platform that is 30 m above a bridge and they're gonna step off that platform, they're gonna come down below the bridge 30 m. OK. So that's that total distance of 60 m that they've fallen. Then that cord is going to stretch that distance of Y until we get to the maximum extension point. And we wanna figure out what that maximum extension Y of the rope is. OK. So we have some gravitational potential energy here because the thrill seeker starting from some height, we have some kinetic energy at some point through this because they're gonna be moving, we have some potential energy to, to spring a spring potential energy because we have this cord, this rope. And so what we want to think about is the conservation of mechanical energy. So if we think about our thrill seeker when they are on the platform, and we're gonna consider the bridge to have a height of zero and that's gonna be our height zero. So when the thrill seeker is at that platform, they're gonna be 30 m above the bridge. And so the initial height is going to be 30 m. Now their initial speed V not, we're gonna consider that to be 0 m per second. So we're gonna consider that they're just kind of falling or stepping off that platform. So their initial speed, initial velocity is just zero. And we know that the cord at this point or this elastic rope is uns stretched. So XK not that stretch of that rope is just going to be 0 m. Ok? So everything at the beginning that we have going on. We've written down, we wanna do the same for when we're at this stretched point in our court. And so at the very bottom of our picture, what do we have going on? Well, our final height, HF OK, if the bridge is height zero, then our final hit is going to be negative because we're going below that bridge, we're taking upwards to be positive. So we have the 30 m that we travel before the rope stretches plus the distance. Why that the rope stretches? So our final height is gonna be negative. And then in brackets, we have 30 m plus y, our final speed is again gonna be 0 m per second just like the initial speed, ok? And the reason for that is that we're at maximum extension of the rope. So this thrill seeker is going to come to rest momentarily as that rope gets to its maximum extension as they go from going downwards to being flung back upwards. And so momentarily at that point, they have no speed and the stretch of the rope at this point is gonna be that maximum extension that we're looking for, which is why, ok. So what we're looking to calculate is why and we're gonna do it using conservation of mechanical energy. So the mechanical energy at the beginning of all of this emac NOTT is gonna be equal to the mechanical energy at the end, emac f what makes up our mechanical energy? Well, we have kinetic energy and potential energy and our potential energy in this case comes from two sources, we have the gravitational potential energy and the elastic potential energy that potential energy due to this cord. So if we write out all of the terms, we have k, not initial kinetic energy plus UG, not initial gravitational potential energy plus us, not the initial elastic potential energy. This is gonna be equal to KF plus EU GF plus us F OK. So all of those corresponding values at the final time point. Now the kinetic energy, we've said that the speed is zero at both the top and the bottom, the initial and final time points. So we have no kinetic energy. And so those K terms are gonna be zero and then initially our rope has not stretched. So we have no elastic potential energy. So us nat is going to be zero as well. So we're taking all of that initial gravitational potential energy and we're converting into, into some gravitational potential energy and some elastic potential energy. When we do that, we're gonna be left with the following equation. So our gravitational potential energy recall is given by MGH. The elastic potential energy is given by one half kx squared. So we get MGH not is gonna be equal to MGHF plus one half KXF squared. All right, let's substitute in our values. We know that each knot is 30 m. And the mass of our thrill seeker is 70 kg. We know the acceleration due to gravity is 9.8 m per second. So we get 70 kg multiplied by 9.8 m per second squared, multiplied by 30 m is going to be equal to again that mass 70 kg multiplied by the acceleration due to gravity 9.8 m per second squared multiplied by that final height. And we are gonna write this as negative 30 m minus one is we've just brought that negative into the brackets. Then we're adding one half multiplied by K that spring constant, which we're told is 50 newtons per meter multiplied by this final stretch, which is the maximum extension, which is this value of Y that we are trying to find and that is squared. OK. So we're looking for Y in our equation, we have a Y squared term, we have a Y term, we have some constant terms. So let's simplify. We're gonna try to combine some terms if we can expand and we're gonna try to write this as a quadratic equation that allows us to use the quadratic formula to solve for. So expanding we get 20,580 kg meters squared per second squared. This is gonna be equal to negative 20,580 kg meters squared per second squared minus 686 kg meter per second squared multiplied by Y plus 25 newtons per meter multiplied by Y squared. We're gonna move everything to one side and we can write this as 25 newtons per meter multiplied by Y squared minus 686 kg meters per second squared multiplied by Y minus 41,160 kg meter squared per second squared. That is all going to be equal to zero. And now that we have it all equal to zero, we can use a quadratic formula. So let's take a look at the equation. We have, we need to figure out what our A B and C values are. So our a value that's a coefficient associated with Y squared. It's gonna be 25 newtons per meter. The B value the coefficient associated with Y is gonna be negative 686 kg meters per second and RC value at constant term negative 41,160 kg meters squared per second squared. Now we're called the quadratic formula. Y is equal to negative B plus or minus the square root V squared minus four AC all divided by two A. We have our A B and C values. All we need to do is substitute the, we get that Y is equal to 686 kilogram meters per second squared. Plosser minus the square root of 686 kg meters per second squared all squared. OK. And this is a negative value. But when we square, it doesn't matter if we take the negative or the positive, just save a little bit of space there. Now we're subtracting four multiplied by a 25 newtons per meter multiplied by C negative 41,160 kg meters squared per second squared. And all of this is divided by two A. So two multiplied by 25 newtons per meter. All right. So this looks really messy, but we can work this out on our calculator. In terms of the unit we have inside the square root, we have kilogram meter per second squared. That's equivalent to a Newton. And then we're squaring it. OK? So that's the first part of the square root. The second part of the square root we have Newton multiplied by kilogram meter squared per second squared divided by meter. OK? Well, that gives us Newton squared again. When we take the square root, we're gonna end up with new ones. OK? In the first term kilogram meter per second squared, again, that's a new. So in the numerator, we have Newton, then we're dividing by newtons per meter. That's gonna give us a final unit of meters, which is what we want. OK? So it's always important to make sure that new units match and then we can do the calculation and we're gonna get two Y values here. So if we add, we're gonna get 56.55 2 7 m. And if we subtract, we're gonna get negative 29.1127 m. Now this second solution, ok? This doesn't make sense. We're talking about a distance, ok? So we want the positive value and so what we're gonna do is take that first value. And so the maximum extension of our rope is going to be about 56.55 m. If we go back to our answer choices, we compare what we found and we round, we can see that the correct answer is going to be option C 56.6 m. Thanks everyone for watching. I hope this video helped you in the next one.

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