At work in Alaska you must pull two heavy boxes of supplies across a frozen lake

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

For the box being pulled by rope,
only force acting is the tension.

therefore, T=Ma where T=tension,M=mass,a=acceleration

now T < 1000N or else the rope breaks
or, Ma < 1000 ( as T=Ma )
or, a < 1000/M
or, a < 1000/500 {M=500kg}
or, a < 2
therefore,maximum acceleration of box without breakage of rope is 2m/sec2.

now, for the box being pulled by cable :
force F acts in the forward direction and tension T acts in the backward direction.
or, F - T =Ma
for maximum F without rope breaking, a is maximum.
or, F - T = M*2
or, F = 2M + T

therefore maximum force that can be applied without breaking the rope :
F = 2*500 + 1000 {M=500,maximum T=1000}
or, F=1000 + 1000
or,F= 2000N

since the force F gradually increases from 0 to 5000N,therefore the rope breaks when F exceeds 2000N.

Also,if the rope breaks then it is quite evident that the second box does not cross the lake.Therefore only one box which was pulled by cable crosses the lake.

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