Pendulum Clocks and Their Errors . l arcs AA or BB. The effect of the escapement forces on the pendulum can be representedgraphically. In fig. 5 let time be measured along OX, and let ABC represent theoscillation of the pendulum {y ?= a sin ^irtlr approximately, where a isthe maintained amplitude). Let y = ± OG- and y = + OH be straightlines parallel to OX distant k and e respectively from OX, whereK =fr/\{ijb ?+ tan/3) and e = jJifr/A. If the scale on which ABC is drawnis such that ordinates represent the gravitational force of restitution aswell as the displacement, these lines, as will be r
Pendulum Clocks and Their Errors . l arcs AA or BB. The effect of the escapement forces on the pendulum can be representedgraphically. In fig. 5 let time be measured along OX, and let ABC represent theoscillation of the pendulum {y ?= a sin ^irtlr approximately, where a isthe maintained amplitude). Let y = ± OG- and y = + OH be straightlines parallel to OX distant k and e respectively from OX, whereK =fr/\{ijb ?+ tan/3) and e = jJifr/A. If the scale on which ABC is drawnis such that ordinates represent the gravitational force of restitution aswell as the displacement, these lines, as will be readily seen, are theequilibrium positions of the pendulum when acted on by the constanthorizontal forces fr/X(fi + tan^S) and fifr/X, and the oscillation, during the 1911.] Pendulum Clocks and their Errors. 513 intervals for which each force lasts, is in reality a harmonic vibration ofperiod r^, but whose virtual amplitude and phase change abruptly with thealteration of the zero position. Thus at Pi the force changes suddenly. A C -Fig. 5. from 3/ — <3 to 2/ + /c, at P2 from 3/ + /c to y + e, at P3 from 3/ + a to ^ — e(on account of the change in direction of the motion), and so on. Thesechanges are indicated by the thickened line. Of course there is no sudden change either in the velocity or positionof the pendulum when the zero positions change, and these two conditions,together with the constancy of the period Ta for all the portions comprisedbetween P1P2, P2P3, of the oscillation, suggest a simple construction forevaluating the effect of the applied forces on the time which elapses betweenthe successive elongations, that is on the effective period of the pendulum. In fig. 6 let OEOE = ±h OGOG + K, OHOH = ±^J K and e having the same meanings as in fig. 5. B \^ X ^—--. / /^ ^^^ Pa / E 7 \^ —-^^^^ \ / H- ^ \ 1, ) G^ ? xT^ \\ // P ??~-^,^ ^?^-^^j^^.-^ \ Xj / V ^^^ Fig. 6. 514 Mr. H. K A. Mallock. [May 27, We may imagine the actual motion of the pendulum
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