. A text book of physics, for the use of students of science and engineering . angle BAD is very small, AC and ADwill be sensibly equal. Let L be the length of thethread, then BD mg F = mg AC BD. (1) Hence we may say that F is proportional to BD for vibrations ofsmall amplitude. BD and BC coincide nearly for such vibrations,and the body will execute simple harmonic vibrations under theaction of a force F which varies as the displacement of B from the To obtain the value of /^, make BD = 1 in (1), vertical through A giving Now mq T = 2 I1 o -v- ImLV mg •(2) XVI VIBRATIONS DIFFERING IN PHASE 225
. A text book of physics, for the use of students of science and engineering . angle BAD is very small, AC and ADwill be sensibly equal. Let L be the length of thethread, then BD mg F = mg AC BD. (1) Hence we may say that F is proportional to BD for vibrations ofsmall amplitude. BD and BC coincide nearly for such vibrations,and the body will execute simple harmonic vibrations under theaction of a force F which varies as the displacement of B from the To obtain the value of /^, make BD = 1 in (1), vertical through A giving Now mq T = 2 I1 o -v- ImLV mg •(2) XVI VIBRATIONS DIFFERING IN PHASE 225 Example.—Find the period of a simple pendulum of length 4 feet ata place when g is 32 feet per second per second. Find also the frequency. 7 >32 2-222 seconds. n = T 2-222 -zr-;^ =0-449 vibration per second. Vibrations differing in phase.—In Fig. 256 (a), two points Px and P2rotate in the circumference of the circle with equal and constantangular velocities. Their projections Mx and M2 on AB executesimple harmonic vibrations which are said to differ in phase. The. Fig. 250.—Vibrations differing in phase. phase difference may be defined as the value of the constant angleP,OP2 = <£, and may be stated in degrees or radians. Thus a phasedifference of 90° or tt/2 radians would give vibrations, such that Mxwould be at the end A of the vibration, at the instant that M2 waspassing through the point O. The vibrations possessed separately by Mx and M2 may be impressedon a single particle, which will then execute simple harmonic vibra-tions compounded of the vibrations possessed by Mx and M2. InFig. 256 (b) construct a parallelogram by making P,P and P2P equaland parallel respectively to OP2 and OPx. Join OP and draw PMperpendicular to BA produced. OM2 and IV^M are equal, since they are the projections on AB ofequal lines equally inclined to AB. Therefore OM is equal to thesum of the component displacements OM1 and OM2. Hence, if theparallelogram OP]PP2 rotate about O wit
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