Synopsis
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. 1900 Excerpt: ...The position of Q at any time is determined by the equation Equations (1) and (2) are valid for all positions of R and Q, since vx changes sign with y, and cos (at) passes through zero at O. 89. The time required for Q to move from A back again to A is especially to be noted; this is equal to the time in which R moves over one entire circumference, reckoned from any starting-point. During this time Q completes one of its cycles, or sets of values for position, velocity, and acceleration. We Tx is termed the period of the vibration; and the number of periods in one second, which is the reciprocal of Tu is the frequency. Another prominent value of the time is that which elapses while Q passes between A and B, or between any two positions corresponding to motion of R over a semicircumferT ence; it is--. We have then ti =--=-=--. (2) The period of a simple harmonic motion is seen to be independent of the amplitude. It is clear that the time of passing once round a circle depends upon the given angular velocity, and not upon the radius. Consequently the constant k determines the period for Q, whatever amplitude may be given to the vibration. Such oscillations are isochronous. The time in which Q moves from any position x' to any other x", is the same as that in which OR moves through the corresponding angle. The relation can be expressed in the form, x"--x, = Xi cos (ot")--cos (ot')'. (3) The values of a;" and x, are equal if t" + t' = nTi; or t"--«' = n2; n being any whole number. Under the first supposition the two positions of Q coincide, but the speeds are equal with opposite signs; the two positions of OR are symmetrically related to OA. In the second case all the elements of the motion are exactly repeated. These results ...
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