Pendulum Clocks and Their Errors . 188,t See Lambs * Hydrodynamics, § 328, p. 328. J See Maxwells * Electricity, vol. 2, p. 338, from which this result may be easilydeduced. 1911.] Pendulum Glochs and their Errors. 509 /3 is the angle of the equiangular spiral of which the motion of the resistedoscillation may be taken as the projection. Or, in other words, ifAa ( = 2iTci^) is the diminution of amplitude which occurs in each com-plete oscillation when the arc is a, then, owing to the extinction, the periodis diminished by Xa [1 — 27r/v^(47r^ 4- /3^)], which, when ^ is small, becomes (12) The e
Pendulum Clocks and Their Errors . 188,t See Lambs * Hydrodynamics, § 328, p. 328. J See Maxwells * Electricity, vol. 2, p. 338, from which this result may be easilydeduced. 1911.] Pendulum Glochs and their Errors. 509 /3 is the angle of the equiangular spiral of which the motion of the resistedoscillation may be taken as the projection. Or, in other words, ifAa ( = 2iTci^) is the diminution of amplitude which occurs in each com-plete oscillation when the arc is a, then, owing to the extinction, the periodis diminished by Xa [1 — 27r/v^(47r^ 4- /3^)], which, when ^ is small, becomes (12) The effect of a constant resistance on an oscillation is to make eachsuccessive are less than the preceding by a constant quantity. The influenceon the period may be gathered from the fact that each time the direction ofmotion changes sign the zero about which the oscillation takes place isshifted in a direction opposite to that of the motion, and by a constantquantity proportional to the resistance. In fig. 3 let ABC represent the. Fia. 3. unresisted oscillation. Then if OE represent the resistance (on such a scalethat the ordinates of ABC represent the positional forces), the retardedoscillation for the half period from A to B will be performed as an unresistedoscillation about the zero axis y = OE, and from B to C as an unresistedoscillation about ^ = — OE, and the arc a is reduced by OE/X. at eachhalf swing. Thus, if the time is measured between successive elongations, theperiod is unaltered by constant friction; but, between successive crossingsof the unresisted zero line, each successive half period is longer than the p^eceOing one ., ^ (X_i). When ... o, e,« ,«.h. the value t/4, the opposing frictional force is equal to the positional forceand the motion ceases. This proposition is useful in some cases, for it shows that the free periodof a pendulum resisted by constant friction can be found, very nearly, bycounting the swings until the motion stops, then if t
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