Analytical mechanics for students of physics and engineering . ? •(-9 (10) x = a cos M + 6i) and ?/ = 6 sin (co£ + 52). 241. Physical Pendulum. — Any rigid body which is free tooscillate under the ad ion of its own weight is called a physicalor a compound pendulum. Let A, Fig. 137, be a rigid bodywhich is tree to oscillate about a horizontal axis through thepoint 0 and perpendicular to the plane of the paper. Fur-ther lo c denote the position of the center of mass and D itsdistance from the axis. Then the torque equation gives /^ = -mgDBmd, (X) where m is the mass of the body and 0 the angular
Analytical mechanics for students of physics and engineering . ? •(-9 (10) x = a cos M + 6i) and ?/ = 6 sin (co£ + 52). 241. Physical Pendulum. — Any rigid body which is free tooscillate under the ad ion of its own weight is called a physicalor a compound pendulum. Let A, Fig. 137, be a rigid bodywhich is tree to oscillate about a horizontal axis through thepoint 0 and perpendicular to the plane of the paper. Fur-ther lo c denote the position of the center of mass and D itsdistance from the axis. Then the torque equation gives /^ = -mgDBmd, (X) where m is the mass of the body and 0 the angular displace-ment from the position of equilibrium. (9) x = a cos [cot — J and y = b cos ( cot + - PERIODIC MOTION 309 The equation —:T = -k-sinx isar not integrable in a finite number of terms; therefore the solution of equation (X) must be given either in an approximate form, or it mustbe expressed as an infinite Approximation. — When6 is small sin 0 may be replaced by6. Therefore we can writerdco. dt mgDd, (X) Fig. 137. dt (X) where mgDI It be observed that the last two equations are of the same type as equations (I) and (II) ofp. 297, the differential equations of simple harmonic the motion of the physical pendulum is approxi-mately harmonic. Hence we can apply to the present prob-lem the results which were obtained in discussing simpleharmonic motion. Thus the expression for the displace- ment i^ a sin (ut+ 8), XI) where a is the amplitude, , the maximum angular dis-placement of the pendulum. On the other hand the periodof the pendulum is 2 v ^o = —r = 2ttv/ \ mgD /Ir+mD* - V —~—> mgD = 2 IK-+D2 XII (XIII) 310 ANALYTICAL MECHANICS where / is the momenl of inertia of the pendulum about anaxis through the center of mass parallel to the axis of vibra-tion and K is the corresponding radius of gyration. Second Approximation. — Starting with the energy equa-tion we haveddV x(D! mgh = mgD (cos 6 — cos a),or dt = I dd 2 mgD VC
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