This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1907 Excerpt: ...of Simple Harmonic Motion, 3, (I), n2 having here the value g/l. Hence when a simple pendulum swings through a small amplitude, its motion is approximately harmonic and its period is approximately 9 A question that interested the mathematicians of the eighteenth century was this: In what curve should a pendulum swing ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1907 Excerpt: ...of Simple Harmonic Motion, 3, (I), n2 having here the value g/l. Hence when a simple pendulum swings through a small amplitude, its motion is approximately harmonic and its period is approximately 9 A question that interested the mathematicians of the eighteenth century was this: In what curve should a pendulum swing in order that the period of oscillation may be absolutely independent of the amplitude? It turns out that the cycloid has this property. For the differential equation of motion is d2s m--=-mgsmr, where s is measured from the lowest point, and since, from Ex. 8, p. 151, s = 4asinT, d's a we nave--=---s. dt1 4 a This is the differential equation of Simple Harmonic Motion, 3, (I), and hence the period of the oscillation: 9 y9. is independent of the amplitude. A cycloid pendulum may be constructed by causing the cord of the pendulum to wind on the evolute of the path. But the resistances due to the stiffness of the cord as it winds up and unwinds would be appreciable. We will close this paragraph with a general theorem. Suppose a bead slides on a smooth wire of any shape whatever. Then its velocity at any point will be the same as what the bead would have acquired in falling freely under the force of gravity the same difference in level. We have already met special cases of this theorem in the inclined plane and the simple pendulum. We shall restrict ourselves to plane curves, but the proof can be extended without difficulty to twisted curves. Newton's Second Law of Motion gives If we suppose the bead to start from rest at A, then 0 = 2gx0] O, But the velocity that a body falling freely a distance of x--x0 attains is expressed by precisely the same formula, and thus the theorem is established. In the more general case that the bead p...
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