# The Differential Wheel and Axle.

Yesterday whilst I was pointificating upon the joys and wonder of collecting beer cans as a hobby I posted a photograph of a small model of a rather complicated member of the ‘Simple Machines’ family.

Now it doesn’t happen often but somebody (Margie) who shall remain nameless hinted indirectly that I could maybe pretty please explain the effect of a windlass have an axle that has two different diameters.

So here goes.

This machine is sometimes called a Chinese Windlass and it was used on the Australian gold fields by Chinese miners. It was very effective.

First a preliminary explanation of the mathematics of a simple lever. Archimedes said about 2350 years ago that if he had a lever long enough he could move the world.

The longer the lever is on the left of the fulcrum and the shorter it is on the right then the more you can lift with the least amount of effort. But the downside is that you have to move your end a long way and you only get to move the earth a small way. I won’t go into the mathematics of it now but there is a diagram at the bottom that any interested parties can peruse.

A wheel is just a continuous collection of levers.

In a simple windlass you have a rope wound around the inner circle (axle) and a handle on the outer circle.The amount of advantage you get (forgetting friction) is determined by how much longer the distance is from the Effort to the Fulcrum than the distance from the Load (Resistance) to the Fulcrum.

Let’s say that the log (axle) is 20 cms across and the wheel is 100cms across then to lift a bucket weighing 5 kgs you only need to apply 1 kgs to the point P. Again I haven’t taken into account the friction of the bearings. But you get the idea.

Let’s have another look at my model.

The big wheel is 50 mm and the axle on the left is 10 mm. Now I will change all measurement to the distance the string travels around the three wheels.

The big wheel is ≈160mm. The big axle is ≈31mm and the small axle is ≈25mm.

If it was just the big axle it would be 160 divide by 31 = ≈5. So on its own I could lift 5kg with an effort of 1kg.

But every time I move the handle around 160mm the big axle winds in 31mm but the small axle lets out 25mm. So it is like I have one wheel with a circumference of 160mm and one axle of circumference 6mm. 160/6 = 26.

But there is more. Because the load is carried on a pulley at the bottom the weight is haved on the string. So everything I have just said is doubled.

And that’s a huge difference.

But that’s not all folks. Because of friction and the string going one way and then the other the machine is not efficient. I won’t go into the mathematics but efficiency is a relationship between the amount of effort required to lift a load and the amount of distance you have to make that effort. In the case of this machine the distance of the effort is very large and the distance of the load is very small but it is worth it because the load is so big.

But there is one other advantage. Let’s go back to the first windlass. If you wind up the bucket out of the well but you let go the bucket will fall back down the well and you will get smacked in the head by the big wheel spinning around at a great rate. In other words it returns to its state of equilibrium.

But when the efficiency ratio of a machine is less than .5 the lever will not return to where it was. It will just stop.

So if you are a miner – for example – and you are using the Chinese windlass and you stop for a smoke then the windlass will stop just where it was. A European using his simple windlass would need to have something to stop the rope unwinding. Officially it is known as ‘overhauling’.

So now do you understand why my class called it the Captain’s Magic Machine and why they made me a model.

## 11 thoughts on “The Differential Wheel and Axle.”

1. This is brilliant! Thanks for explaining the magic machine so eloquently. I was pretty bad at physics, but it fascinates me in small doses. I can see how this model would be something to treasure–it is kind of symbolic and akin to the difference in the impact a good teacher can make…

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2. I failed every single science exam at ‘o’ level. At the highest level Physics, Chemistry and Biology. Next step down Chemistry with Physics and Physics with Chemistry and right at the end of the scale (to my shame) General Science. No one failed General Science except me. I remember being genuinely shocked. I blame a mutant algorithm!

I went on to study History and English.

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3. Sorry I did not make it to the end of the post but your love for Physics and teaching shines through.

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