I am fascinated by the idea of a tug of war between Dover, UK, and Calais, France, with thousands of people on each side, 42 kilometres apart. Could a rope be made long, light and strong enough for this?
Martin Gellender
Brisbane, Australia
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The maximum distance that can be spanned by a rope or cable is limited by it sagging under its own weight.
Near the centre of the span, the cable will only be slightly curved and tensile force within the cable will be nearly constant. But away from the centre, the upwards slope of the cable becomes steeper and the tensile force increases.
The sag of the cable can be reduced by increasing the tensile force 鈥� by pulling harder at the ends 鈥� but only up to the tensile strength of the material. The tensile stress could be reduced by increasing the thickness of the cable, but this adds to its weight, so there is still a maximum span.
I calculate that the maximum distance that can be spanned by a steel cable with a tensile strength of 1000 megapascals is 40 kilometres. So, a tug of war with such a cable across a distance of 42 km would be impossible. The maximum span could be increased by using materials that are lighter and stronger than steel, such as titanium alloys, although such materials tend to be much more expensive than steel and more difficult to fabricate.
Eric Kvaalen
Les Essarts-le-Roi, France
The problem is that the rope would go down into the water unless the tension were many times its weight.
A rope made of the synthetic fibre Kevlar, for instance, can support about 250 kilometres of itself if held vertically. But in the tug of war, it would be in the form of a catenary (a rope suspended between two points at the same height). Even if the people on the English side are standing on top of the cliffs of Dover, a Kevlar rope would be too weak to keep itself from touching the water.
Garry Trethewey
Cherryville, South Australia
This tug of war would be possible via the Channel Tunnel. It would also be possible with a rope floating on the sea 鈥� though this might entail other problems.
In round figures, let鈥檚 say there are 10,000 people on each side, each weighing 100 kilograms. Let鈥檚 also say that each can lean back 45 degrees before they slip, and so they can each pull with 100 kg of force.
Functionally, the number on one side would act like an anchor to the number on the other side, so we wouldn鈥檛 have to count all 20,000 people, just the ones on the side where the anchor fails. We now have 10,000 脳 100 kg, or 1000 tonnes, pulling on the rope.
If it were made from Dyneema, which is very strong and so light that it floats, I calculate that a rope with a 1000-tonne breaking strain would have to be 120 millimetres in diameter. But this would only be good for a normal 鈥渇ace-to-face鈥� tug of war. Next we need to factor in a 42-kilometre separation.
Would the friction from 42 km of rope make a difference? In a tunnel, yes. This rope would weigh just under 300 tonnes and there would be some friction between the rope and the tunnel floor. How much? It is hard to put numbers to this, but imagine dragging a rope along the ground. Tested on my floor, my lawn and some rough ground, the force needed to drag the rope is less than the weight of the rope. With Dyneema being fairly slippery and the floor of the tunnel being smooth, this comes in at less than the 300 tonnes, so wouldn鈥檛 materially alter the situation. If, alternatively, our tunnel were 4200 km long, the rope would weigh 30,000 tonnes and neither side could move it.
If the rope were floated across the channel in still water, friction would be nil, but a strong tide might drag the rope and pull both teams into the water.
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