C.O.M. Study Guide

A common misconception in dealing with C.O.M. problems is misunderstanding the standard formula: 
∑x_{i}m_{i} + ∑y_{i}m_{i} + ∑z_{i}m_{i}

∑m_{i} 
I always suggest that when you have a problem that isn't making much sense, make is simple as you possibly can. So... the best way to simplify this situation is to evaluate the C.O.M in only one dimension: 
∑x_{i}m_{i} ∑m_{i} 
It's *REALLY* easy to lose the forrest through the trees on this one if we consider x_{i} to be a measure of the length of one of the objects... NEGATIVE!!! x_{i} is a measure of the distance to the C.O.M. of that object!! Let's do a for example: Consider the case of a modern shipping container with the dimension of length = 6.2 m and width = 4.1 meters and height = 2.6 meters and a mass of 9000 kg (let's keep it simple and ignore height for the moment). If we simply plugged in numbers to find the C.O.M of that object we would get something like: 
x_{i} = 6.2 m, y_{i} = 4.1 m, z_{i} = 2.6 m and m_{i} = 9000 kg. 
But we're going to keep it simple and only find the C.O.M. in the x dimension. sooo... using our formula we SHOULD be able to calculate the center of mass in the x dimension for that shipping container: 
∑x_{i}m_{i}
∑m_{i} 
substituting our values we get: (6.2 m)(9000 kg)/(9000 kg) 
= 6.2 m ==> which is CLEARLY not the center of mass in the x direction... what the heck is going on here? Recall that x_{i} is NOT a measure of the length of the shipping container but rather a measure of the distance from some reference point (usually the origin) to the CENTER OF MASS of that shipping container. SOOOOOO... Imagine the container is sitting on an xy coordinate graphic system with the corner at the origin. Let's try again: If the shipping container is 6.2 m in length, then the C.O.M. of that container in the x direction is obviously 3.1 m... SO: 
∑x_{i}m_{i}  ∑m_{i} 
(3.1 m)(9000 kg)/(9000 kg) 
which is simply 3.1 m in the x dimension This may seem absurdly easy, but remember, the emphasis is on measuring the distance to the center of mass, not the length of the object in quesiton. TREAT OBJECTS AS POINT SOURCES WITH EACH POINT AS THE C.O.M. of that object! 
EXTERNAL LINKS: Hyperphysics 