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Now we know both components of the velocity at the point N 1 One in the direction ol` ON and equal to v cm/sec, the other perpendicular to it and equal to it >< ON/30 cm/sec. All we have to do is to add them according to the parallelogram law. The diagonal determines the velocity of the combined motion and at the same time shows th6 direction of the tangent NT to the spiral at any given P0int. 23. The Chain of Galilei In his book "Dialogues on Two New Sciences", which was lim Published in Italian in Leiden (the Netherlands) in 1638, Galilei si1g8€Sted the following method of constructing a parabola: "Drive N0 113ilS into a wall at a convenient height and at the same level; make the distance between them twice the width of the rectangle upon which it is desired to trace the semiparabola.

Consider the following question: what form should be given to a well—polished metal trough connecting two given points A and B (Fig. 39) so that a polished metal ball rolls along this trough from point A to point B in the shortest possible time? At first it seems that 34 one should choose a straight trough, as it provides the shortest way from A to B. But we are trying to find the shortest time and not the shortest way, and the time depends not only on the path itself but on the speed of the ball as well.

Imagine that the curve is drawn on a perfectly vertical, polished wall and that we can drive in 41 nails at different points of the curve. Let us drive them in at points A and B that are at the same level, as Galilei suggested (this condition is not essential, however). Choose a light chain whose length is exactly 2I, the length of the arc AB, and fix its ends at A and B. The chain will hang in the form of the arc shown above. There will be no gaps between the hanging chain and the curve. Yl A 7:3-le + e ) B l x` /’ \\ C / i [ <‘ E Od* Figure 49 The selection of a chain of the required length can be done by trial and error.

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