Subtle Saturday Science of String

The subtle science of pieces of string…

Moving Very Heavy Things With a Rope & Not Much Else: ‘Springing’ #springing
Lewis Dowels: Gripping Giant Blocks of Stone: Recommended for Pyramid Builders #pickblock
Balloon Biceps: making muscles out of balloons and string #balloonbiceps

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Moving Heavy Things by ‘Springing’

The geometry of stretched strings and ropes offers a surprisingly easy way to move heavy objects around. Not only is the method easy on the muscles and easy on the apparatus needed, it is easy on the brain: the application of some trigonometry – just resolving forces at a point – is all that is required to analyse what is going on.

What you do

Suppose that you have heavy lump M, which exerts a force Mg on the ground, and has a frictional force equal to half its weight, i.e. 1/2 Mg. Maybe M is a piano weighing in at 250kg, the castors of which are hopelessly corroded and act simply as skids. Now you could just tie a rope around pull, but you would have to pull with a force of at least 1/2 Mg, which is 1250N, 125kg force: that’s hard ! But nil desperandum, do not despair…

Now suppose you tie the rope off on fixed point A and then wrap it around M. Maybe A is a heavy gauge anchor point in the wall. Make sure that the rope is pulled tight. Perhaps an assistant can hold the end of the rope to stop it slipping around M. Now pull on the mid-point of the free length of rope. A surprisingly small force, just a couple of hundred N, ten or twenty kg f, and the piano M will shift a little. But it doesn’t move too far. So if your assistant simply pulls the wrap on M tight again, you can repeat the exercise.

How does this work ? The Science and the Math

Take a look at the geometry in the diagram. You are pulling at right angles to the rope with force f, and this is exerting a much large force F. The rope is pulled sideways by a distance D, producing an angle of sin-1(D/(L/2) or sin-1(2D/L). Looking at the balance of vertical forces, the vertical component of the force F produced by the load M and the fixed point F are the same and are additive: So 2 F sinA = f. If angles are small and that sinA ~ A in radians, 2 F A = f, and A = 2D/L. Now F / f = 1/(2 sinA) ~ 1/(2 A) = 1/(2 2 D/L) = 4 L/D.

So the force you applied is boosted by a factor of B, given approximately by a quarter of the free length L of the rope, divided by the distance D that you have pulled the rope from its original straight line, or B = L / 4 D . With a deflection D of say, 100mm on a length of rope say, 4000mm, B = 4000 / 400 = 10x.

References: pages 45-50″String Nutcracker” in Neil A Downie Vacuum Bazookas, Princeton

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Lewis Dowels: Picking Up Giant Blocks for Pyramids

The gripping tale of how the geometry of stretched ropes again comes to the rescue when us Pharoahs, or anyone else, wants to pick up blocks of stone and cement them into Brobdingnagian Pyramids.

What you do: Take a large block of wood, say 1kg to 5kg, and two short length of round wooden dowel rod, say 5cm to 8cm long, 12mm diameter. Drill two small holes near the of the dowel rods big enough for strong string to be threaded through. Now drill two holes vertically in the block, symmetrically either side of the centre of the block 5 or 10cm apart. They should be just large enough for the round dowels to be pushed in, deep enough for just their last couple of cm to be sticking out.

Now put the dowels into the big block and tie a loop of string around the two, via the holes, making the loop fit reasonably tightly, as shown. Now use a lifting string with a loop on the end placed as shown on the middle of the loop around the dowels, and pull up. Surprisingly, instead of pulling the dowels out, the lifting string pulls the loop up and pulls the two dowels together, making them grip in the holes in the block, and allowing it to be lifted. Give the lifting string a shake, to see if it will loosen. You will find the arrangement quite resistance to ‘being fooled into’ letting go of the block.

Why it works, the science and maths

The science and the maths of Lewis dowels depends upon the geometry of springing, similar to that seen above in moving heavy loads. But there is, in addition, the laws of friction. Once the rope pulls upwards, it pulls sideways, as in the diagram, and the pull on the rope is magnified by the factor B = L/(4D) a seen with rope springing. But here that force is magnifying the force which applies to the dowels to produce friction. Both the dowels contribute, so the force is doubled, and each is multiplied by the coefficient of friction u, often in the range 0.2 to 0.5 .

An examination of how they fit will convince you that only the inside of the top of the dowels grip hard, the bottoms don’t press very hard. However, it is only a doubling, not a quadrupling, of the friction force. So what is the minimum coefficient of friction for Lewis dowels to work ?

Now f = M g, where M is the mass of the stone being lifted, and f is multiplied by L/4D, so friction force = 2 u (M g . L/4D ). But if the weight of the block exceeds the friction force, the block will slip:

If M g > 2 u (M g . L/.4D) OR if 2 u L /4D <1, then slipping will occur.

Doing some calculations with typical numbers, however, the stone won’t slip: If u is 0.5, L 200mm, with D 20mm, then 2 u L / 4D has the value 200/80, or 2.5, giving a reasonable margin of safety.

And finally… There are many variations you can make. You could, for example, use a different number of dowel rods (going down to one dowel is’n’t, of course, an option !). The dowels don’t have to have holes in them, the string could be fitted to the dowels in other ways, perhaps small screws or bolts, or a ring-shaped groove. And how big can you go ? Does the same effect work in larger sizes ? The clue is above, of course, very large stones were lifted with similar approaches in ancient civilisations. Below is a stone of mass over 40kg, which was lifted with the rope hanging from the lever, allowing the precise positioning of the stone between its very close neighbouring blocks. This would be impossible with a rope simply wrapped around the stone.

Why Lewis Dowels ? Lewis Dowels seems like a good name for this method of lifting large block, because a ‘lewis’ or ‘lewisson’ is the name of a metal gadget that is used in a similar way. Take a look at https://en.wikipedia.org/wiki/Lewis_(lifting_appliance)

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Balloon Biceps

More coming soon…

Balloons are complicated. They look so simple. A simple bag of rubber which you can inflate with air or another gas. But they are made, not of a standard kind of flexible sheet material, but from an elastomer. Instead of simply going from flat to round, the latex rubber of which they are made also stretches – enormously – and the behaviour as this happens is utterly different. The rubber stretches by a factor of 5 or more, so much that it becomes obviously thin, almost transparent, depending upon the colour. The volume of air contained inside the balloon can increase by the cube of the linear stretch: 125x ! The volume of rubber itself is almost constant, so the thickness has to go down to compensate for the area stretching as the square of the linear expansion.

But the fun doesn’t stop there. Put that balloon inside a string net, stretch those net strings between two points on two limbs, and you have a muscle.