At work in Alaska you must pull two heavy boxes of supplies across a frozen lake
ID: 2047652 • Letter: A
Question
At work in Alaska you must pull two heavy boxes of supplies across a frozen lake to your camp.Fortunately there is a machine at your camp with a long cable attached that can pull the boxes for
you, so you attach the cable to one box and then tie a rope between that box and the other one. The
boxes are heavy, each having a mass of M = 500 kg but they are on skis that have negligible friction
with the icy lake. The machine can exert any force F you like from 0 N up to 5, 000 N, but the
rope will break if the tension exceeds T = 1, 000 N. When you turn the machine on you gradually
raise F from 0 toward its maximum (that is what I would do, anyway), not thinking about the rope
between the boxes. The general question is “what happens?” The approximations you should make
are neglect friction, neglect the mass of the rope, and notice that when the rope is taut then the
acceleration of the two boxes is the same.
1) Set up an equation for the maximum force that the machine can exert without the rope breaking,
using only the symbols F, T , and M . (Suggestion: draw a free-body diagram for the box that is
being pulled by the rope. That will give a formula for the maximum acceleration of that box.)
2) Does the rope break?
3) How many boxes (if any) get across the lake?
Explanation / Answer
Let us assume the box attached to the cable is box A, and the one only attached to rope is box B. 1. Set up equations for the limit condition, when the rope is just about to break (if it will). For box A, 2 forces acting on it, pulling force by machine, F, and tension by the rope connecting A and B. Obviously these 2 forces acting in opposite directions. For box B, there is only tension. Then we have F-T=Ma T=Ma When the rope is just about to break, T=1000 N as given. Then from our equation, we can get a=T/M=1000/500=2 m/s^2 Let's put a=2 and T= 1000 back to F-T=Ma, Then we have F= Ma + T= 2000 N So now we can see, when F> 2000 N, no doubt the rope is going to break. And from the question, the max force the machine can give is 5000 N, which is over 2000 N. Thus the rope is going to break if the lake is big enough that the boxes can reach an acceleration of 2 m/s^2. 2. What will happen exactly? Will any box actually cross the lake? Yes, both of the boxes will cross the lake, even though the way you operate the machine will cause the rope to break. Because we consider there is no friction, even after the rope break when the tension of the rope is over 2000 N, box B will keep moving with the same speed and direction as it is when the rope break. So it will eventually cross the lake. On the other hand, box A will still keep accelerate after rope break, as the F on it still exist.
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