Applied Mathematics by Example: Theory by Jeremy Pickles

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In the diagram below, m1 × x1 = m2 × x2 . x1 x2 This Archimedes deduced from simple arguments of symmetry. Suppose, he said, that the masses are measured in some common unit. If the unit is m, suppose for the sake of illustration, m1 = 3m and m2 = 2m. Then the total mass is 5m, and the beam will be balanced if its length is divided into 5 equal parts with a unit mass placed at the centre of each. 63 Rigid bodies Applied Mathematics by Example: Theory Now, without disturbing the overall balance, the three masses on the left hand side can be brought together at their common centre of gravity, as can the two masses on the right: Here it is seen that the mass 3m on the left is positioned one unit of length from the fulcrum, while the mass 2m on the right is at a distance of one and half units, confirming the constancy of the product, mass × distance.

2 The lever The law of the lever was formulated by Archimedes, generally reckoned, along with Newton and Gauss, as one of the three greatest mathematicians in history. Born in 287 BC, he lived in the town of Syracuse in Sicily, and was famous both as a mathematician and as an inventor of mechanical devices and engines of war. Of his work it was said, it is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. . no amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you could have discovered it.

Literally, the term implies a body of zero size, and this would have been a very natural picture for Newton to have in mind as he visualized the motion of a planet through the vastness of space. But more generally, a ‘particle’ means any body, like the falling apple, or Mr B standing in the train, whose size, shape, or orientation are not considered relevant to the calculation. The only attributes of a particle are its position, velocity, acceleration and mass. Often – but not always – this is a useful simplification.

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