Analytical mechanics for students of physics and engineering . 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. Fig. 82. PROBLEMS. 1. Find the center of mass of aright circular c
Analytical mechanics for students of physics and engineering . 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. Fig. 82. PROBLEMS. 1. Find the center of mass of aright circular cone, the density of whichvaries inversely as the distance from the vertex. 2. Find the center of mass of a circular plate, the density of whichvaries as the distance from a point on the circumference. 3. Find the center of mass of a cylinder, the density of which varieswith the nth power of the distance from one base. 4. Find the center of mass of a quadranl of an ellipsoid. 5. Find the center of mass of a hemisphere, the density of which variesas the distance from the center. 119. Center of Mass of a Number of Bodies. Lei mi, rn:,etc., be the masses and xu .<.. etc., be the x-coordinates of tin-centers of mass of the individual bodies. Then if x denotesthe z-coordinate of the center of ma— of the entire systemwe can write x = xdm 150 Similarly ANALYTICAL MECHANICS JfV»i f™2xdm-\- I xdm+ ?o Jo dm+ I dm+ ? • o Jo _ m&\ + m2^2 + • • • mi + m2+ • • •_ HmXj2m 2m 1/// Therefore the mass of each
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