Analytical mechanics for students of physics and engineering . soid. 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 bod
Analytical mechanics for students of physics and engineering . soid. 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 body may be considered as beingconcentrated at its center of mass. ILLUSTRATIVE EXAMPLE. Find the center of mass of the plate indicated by the shaded part ofFig. 83. (a) Suppose the plate to be separatedinto two parts by the dotted line. Thenthe coordinates of the center of mass ofthe lmver part are b , - b — a Xi = - and i/i = 2 2 On the other hand the coordinates of thecciiicr of mass of the upper part are 2b -a b — a 2 and y2 =. Therefore the coordinates of the center of mass of the entire plate havethe following values: b b — a »2 + w* — mi + m»oh(h-a)-+<ra(b — a) —^ ab (6—o)+ff (b — a) a b- + nb - q2(a + 6) 2 + m, 26 TO] + I» 2 j n .b — a. ,, ,2 b —aab{b-a)——\-aa(b-a) —— ab (b — a)-\-aa {b — a)b- + nb - a22 (a+ 6) CENTER OF MASS AND MOMENT OF 151 (b) Suppose the square OA to represenl a plate of positive mathe square OA to represent a plate of negative mass. Then if the twoplates have the same thickness and density the positive and the mmasses annul each other in the square OA. Therefore the two Bquareplates form a system which is equivalent to the actual plate representedby the shaded area of the figure. Bence I he center of mass of the Bquareplates is also the center of mass of the given plate. The masses of the square plates are air and - an-, while the codrdi-nates of their centers of mass are -, -, b , -„
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