Analytical mechanics for students of physics and engineering . 148 ANALYTICAL MECHANICS ILLUSTRATIVE EXAMPLES. 1. Find the center of mass of an octant of a homogeneous sphere,(a) Suppose the bounding surfaces to be x2 + if + z*- = a-, x = 0, y = 0, and z = the limits of integration are x = 0 and x = a. Therefore y = 0 and y = Vo* — x1,z = 0 and z = y/a- — x2 — \ j x c/x d# dz j _ ^o •o Jo f f f dxdyda •O •O -^ 0 = 3a * 8 and by symmetry y = z = — ? 8 (b) Suppose the equations of thebounding surfaces to be given in spheri-cal coordinates, then we have -, 0 = 0, and = - The limits of inte

---

Analytical mechanics for students of physics and engineering . 148 ANALYTICAL MECHANICS ILLUSTRATIVE EXAMPLES. 1. Find the center of mass of an octant of a homogeneous sphere,(a) Suppose the bounding surfaces to be x2 + if + z*- = a-, x = 0, y = 0, and z = the limits of integration are x = 0 and x = a. Therefore y = 0 and y = Vo* — x1,z = 0 and z = y/a- — x2 — \ j x c/x d# dz j _ ^o •o Jo f f f dxdyda •O •O -^ 0 = 3a * 8 and by symmetry y = z = — ? 8 (b) Suppose the equations of thebounding surfaces to be given in spheri-cal coordinates, then we have -, 0 = 0, and = - The limits of integration are r = 0 and r = a0 = 0 and 6 = ?. Fig. 81. 0 and C C f V sin2 0 cos0 rir rid dTherefore x = °x ny ° [x = r sin 6 cos0] f f Cr-smdrirridri<p Jo Jo Jo 8 CENTER OF MASS AND MOMENT OF INERTIA 149 2. Find the center of mass of a righl circular cone whose densityvaries inversely as the square of the distance from the apex, the distance being measured along the axis. dm — t • iry2 • dx T\ Ti7T<7- dx, where n is the density at a unitdistance from the apex. -There-fore Tjira- rhi J. as? f c/x h2

---

Size: 1612px × 1550px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No

Keywords: ., bookcentury1900, bookdecade1910, bookpublishernewyo, bookyear1913