Analytical mechanics for students of physics and engineering . (I) is tho displacement - time curve (II) is the velocity-time curve Fro. L32. I Lb evident from these expressions that the velocity is aBimple harmonic function of the time, that it has the same PERIODIC MOTION 301 period as the displacement, and thai it differs in phase Prom Pthe latter by , as shown in Fig. 236. Energy of the Particle,further explanation. The following do not need -X Fdx 2 w-m 2 iv-m , Of. P2 X 2Tr2a-mP2 sin2 ^(t + to). (VII) e=t+v p 2ir-a2mP2 2ir-a-m (a2 - x2) P2 (*+4).(vi) (VIII) Thus the total energy of
Analytical mechanics for students of physics and engineering . (I) is tho displacement - time curve (II) is the velocity-time curve Fro. L32. I Lb evident from these expressions that the velocity is aBimple harmonic function of the time, that it has the same PERIODIC MOTION 301 period as the displacement, and thai it differs in phase Prom Pthe latter by , as shown in Fig. 236. Energy of the Particle,further explanation. The following do not need -X Fdx 2 w-m 2 iv-m , Of. P2 X 2Tr2a-mP2 sin2 ^(t + to). (VII) e=t+v p 2ir-a2mP2 2ir-a-m (a2 - x2) P2 (*+4).(vi) (VIII) Thus the total energy of the particle is constant and equalsthe maximum values of the potential and kinetic total energy varies, evidently, directly as the square ofthe amplitude and inversely as the square of the period. In Fig. 133, T, U, and V are plotted as ordinates and thetime as abscissa, with phase relations which correspond tothe curves of Fig. (I) is the Uand t Curve. (II) is the T and t Curve.(Ill)isthe Eand tCurve. Fig. 133. 237. Average Value of the Potential Energy. Since I maybe considered as a function of either x or t, we will find itsaverage value with respect to both variables. TakingO and 302 ANALYTICAL MECHANICS a as the limits of x and the corresponding values of t as the limits of t we have p 1 Ca i /•> U*= / Udx 77<=_i_j Udt* a-OJo f-o° =-JrPx*dx 4 p 2 a Jo = ±- firing* _^r P P2 Jo P - 6 ... E=sux=2 Ut- (IX) PROBLEMS. 1. A particle which describes a simple harmonic motion has a periodof.» Bee. and an amplitude of 30 cm. Find its maximum velocity and itsmaximum acceleration. 2. When a load of mass m is suspended from a helical spring of length Land of negligible mass an extension equal to D is produced. The load ispulled down through a distance a from its position of equilibrium and thensel free. Find the period and the amplitude of the vibration. Hookeslaw holds true. 3. Within the earth the gravita
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