Analytical mechanics for students of physics and engineering . (XXI) where Ax = acx. There are three special cases which mustbe discussed separately. Case I. Let o2 = b\ then 6 = 0 for all values of the this is a case of no motion. Casi II. Let b2, then Va2- 62 is by c we have d=Al[e-{a The character of the motionis brought out by the graphof ((inutioii (XXII), 1.). The graph is easilyobtained by drawing thedotted curves, which areplotted by considering theterms of the right-handmember of equation (XXII)separately, and then addingthem geometrically. It isevide
Analytical mechanics for students of physics and engineering . (XXI) where Ax = acx. There are three special cases which mustbe discussed separately. Case I. Let o2 = b\ then 6 = 0 for all values of the this is a case of no motion. Casi II. Let b2, then Va2- 62 is by c we have d=Al[e-{a The character of the motionis brought out by the graphof ((inutioii (XXII), 1.). The graph is easilyobtained by drawing thedotted curves, which areplotted by considering theterms of the right-handmember of equation (XXII)separately, and then addingthem geometrically. It isevident from the curve thatt he value of 9 starts at zero,increases to a maximum, Denoting this (XXII). Fig. then diminishes to zero asymptotically. In this case the motion is said to be aperi-odic «»f dead-bi at. III. a-<b- then \ - is imaginary. Lets i i and V&» - a2 = «. Then vV - 62 = iu. Makingtin- substitution in equation (XXI) we obtaine=Ale-at(eikt-c-ikt)= Axe~al -2 /sin ul*= Ac-a,smut, (XXIII) * See Appendix Avn. PERIODIC MOTION :\X\ where A = 2iAi. Equation (XXIII) is the integral equa-tion of harmonic motion with the additional factor e~at,which is called the damping factor. On account of this factorthe amplitude of the motion continually diminishes. It is evident from equation (XXIII) that the motion isperiodic and has a period V62 - a2 (XXIV) The character of the motion is brought out clearly by the dis-placement-time curve of Fig. 146. A mental picture of the
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Keywords: ., bookcentury1900, bookdecade1910, bookpublishernewyo, bookyear1913
