or the body of the cyclist is moving with speed V with respect to the ground. But look more carefully at what is happening at different points of the spinning wheel. If the wheel doesn’t skid, then the speed of the point of the wheel in contact with the ground must be zero. If the wheel has radius R and is rotating with constant angular velocity Ω revolutions per second, then the speed of the contact point can also be written as V – R Ω. This must be zero and therefore V equals R Ω. The forward speed of the centre of the wheel is V, but the forward speed of the top of the wheel is the sum of V and the rotational speed. This equals V + R Ω and is therefore equal to 2V. If a camera determines the speed of an approaching or receding bicycle by measuring the speed of the top of the wheel, then it will register a speed twice as large as the cyclist is moving. An interesting one for m’learned friends perhaps, but I recommend you have a good pair of mudguards.
7
A Sense of Proportion
You can only find truth with logic if you have already found truth without it.
G.K. Chesterton
As you get bigger, you get stronger. We see all sorts of examples of the growth of strength with size in the world around us. The superior strength of heavier boxers, wrestlers and weightlifters is acknowledged by the need to grade competitions by the weight of the participants. But how fast does strength grow with increasing weight or size? Can it keep pace? After all, a small kitten can hold its spiky little tail bolt upright, yet its much bigger mother cannot: her tail bends over under its own weight.
Simple examples can be very illuminating. Take a short breadstick and snap it in half. Now do the same with a much longer one. If you grasped it at the same distance from the snapping point each time you will find that it is no harder to break the long stick than to break the short one. A little reflection shows why this should be so. The stick breaks along a slice through the breadstick. All the action happens there: a thin sheet of molecular bonds in the breadstick is broken and it snaps. The rest of the breadstick is irrelevant. If it were a hundred metres long it wouldn’t make it any harder to break that thin slice of bonds at one point along its length. The strength of the breadstick is given by the number of molecular bonds that have to be broken across its cross-sectional area . The bigger that area, the more bonds that need to be broken and the stronger the breadstick. So strength is proportional to cross-sectional area, which is usually proportional to some measure of its diameter squared.
Everyday things like breadsticks and weightlifters have a constant density that is just determined by the average density of the atoms that compose them. But density is proportional to mass divided by volume, which is mass divided by the cube of size. Sitting here on the Earth’s surface, mass is proportional to weight, and so we expect the simple proportionality ‘law’ that for fairly spherical objects
(Strength) 3 ∝ (weight) 2
This simple rule of thumb allows us to understand all sorts of things. The ratio of strength to weight is seen to fall as Strength/Weight ∝ (weight) -⅓ ∝ 1/(size) . So as you grow bigger, your strength does not keep pace with your increasing weight. If all your dimensions expanded uniformly in size, you would eventually be too heavy for your bones to support and you would break. This is why there is a maximum size for land-going structures made of atoms and molecules, whether they are dinosaurs, trees or buildings. Scale them up in shape and size and eventually they will grow so big that their weight is sufficient to sever the molecular bonds at their base and they will collapse under their own weight.
We started by mentioning some sports where the advantage of size and weight is so dramatic that competitors are divided into different classes according to their bodyweight. Our ‘law’ predicts that we
7
A Sense of Proportion
You can only find truth with logic if you have already found truth without it.
G.K. Chesterton
As you get bigger, you get stronger. We see all sorts of examples of the growth of strength with size in the world around us. The superior strength of heavier boxers, wrestlers and weightlifters is acknowledged by the need to grade competitions by the weight of the participants. But how fast does strength grow with increasing weight or size? Can it keep pace? After all, a small kitten can hold its spiky little tail bolt upright, yet its much bigger mother cannot: her tail bends over under its own weight.
Simple examples can be very illuminating. Take a short breadstick and snap it in half. Now do the same with a much longer one. If you grasped it at the same distance from the snapping point each time you will find that it is no harder to break the long stick than to break the short one. A little reflection shows why this should be so. The stick breaks along a slice through the breadstick. All the action happens there: a thin sheet of molecular bonds in the breadstick is broken and it snaps. The rest of the breadstick is irrelevant. If it were a hundred metres long it wouldn’t make it any harder to break that thin slice of bonds at one point along its length. The strength of the breadstick is given by the number of molecular bonds that have to be broken across its cross-sectional area . The bigger that area, the more bonds that need to be broken and the stronger the breadstick. So strength is proportional to cross-sectional area, which is usually proportional to some measure of its diameter squared.
Everyday things like breadsticks and weightlifters have a constant density that is just determined by the average density of the atoms that compose them. But density is proportional to mass divided by volume, which is mass divided by the cube of size. Sitting here on the Earth’s surface, mass is proportional to weight, and so we expect the simple proportionality ‘law’ that for fairly spherical objects
(Strength) 3 ∝ (weight) 2
This simple rule of thumb allows us to understand all sorts of things. The ratio of strength to weight is seen to fall as Strength/Weight ∝ (weight) -⅓ ∝ 1/(size) . So as you grow bigger, your strength does not keep pace with your increasing weight. If all your dimensions expanded uniformly in size, you would eventually be too heavy for your bones to support and you would break. This is why there is a maximum size for land-going structures made of atoms and molecules, whether they are dinosaurs, trees or buildings. Scale them up in shape and size and eventually they will grow so big that their weight is sufficient to sever the molecular bonds at their base and they will collapse under their own weight.
We started by mentioning some sports where the advantage of size and weight is so dramatic that competitors are divided into different classes according to their bodyweight. Our ‘law’ predicts that we
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