Three cubes, of side l0 , 2l0 , and 3l0 , are placed next to one another (in contact) with their centers along a straight line as shown in fig. 7–38. what is the position, along this line, of the cm of this system? assume the cubes are made of the same uniform material.

The CM is 61.66568509 units away from the center of the 310 cube in the direction of the center of the 210 cube. First, let's model each cube as a single point around it's center with all of the mass concentrated there. For convenience, I'll place each center on the x-axis with the center of the 310 cube at 0. So we have: For the 210 cube, it's center will be at the sum of half the length of the 310 cube's side plus half of the 210 cube's side. So x = 310/2 + 210/2 = 155+105 = 260. For the 10 cube. The location will be half of the 210 cube's side plus half of the 10 cube's side, plus the center of the 210 cube. So x = 260 + 210/2 + 10/2 = 260 + 105 + 5 = 370 x = 0 Mass 310*310*310 = 29791000 x = 260 Mass 210*210*210 = 9261000 x = 370 Mass 10*10*10 = 1000 Now the center of mass for a collection of objects is CM = sum(Mi*Ri)/M where CM = Location of center of mass Mi = Mass for object i Ri = Coordinates for object i M = Total mass of all objects So let's plug in the values and determine where the center of mass is: CM=(0*29791000 + 260*9261000 + 370*1000)/(29791000 + 9261000 + 1000) CM=(0 + 2407860000 + 370000)/39053000 CM=2408230000/39053000 CM=61.66568509 So the center of mass of the 3 cubes is 61.66568509 units from the center of the 310 cube towards the 210 cube. To demonstrate that the actual location doesn't matter for where the 310 cube is located, let's repeat using 155 as the cube's center location. So all three cubes get shifted over by 155 units, giving centers of 155, 415, and 525. So: CM=(155*29791000 + 415*9261000 + 525*1000)/(29791000 + 9261000 + 1000) CM=(4617605000 + 3843315000 + 525000)/39053000 CM=8461445000/39053000 CM=216.6656851 And if you subtract the location of the 310 cube, you get 216.6656851 - 155 = 61.66568509 which is in the same relative location from the center of the 310 cube.