The center of mass of any given object is the point at which its weight is exerted. With respect to torques and rigid bodies the CM is extremely important. It is the point at which the weight of a rigid body is considered to act. This is extremely important when dealing with torques as the distance the CM (center of mass) is from the pivot point influences the value of the torque.
When an object is symetrical, an object's CM is generally located at its symetrical center. This point may not always be inside the material itself, it may be outside the material as in the case of a Compact Disk, where the CM would be the hole in the middle of it.
While an object's CM doesn't change so long as the object itself doesn't change (if you break the object in half, for instance, the CM will change), the CM of a system can change depending on where the weight in the system is acting. Yes, entire systems have there own CM, not just individual objects. Take a simple straight beam for instance with its CM in the center. If you add a weight to either side of the system, its center of mass changes. Say if you have a beam and add a weight to the right side of it, the CM will displace to the right. This makes sense as the weight would be acting more towards the weight now, or between the original CM and the new weight. It makes sense that the weight of the entire system would now act between the two weight values. From this basic idea (as well as from torque principles), we get the following formula to calculate the CM of any given system.
Using the same principle as we used in any beam problem, we can calculate the CM of a rigid body. The CM equation is a weighted average, like the one used to calculate Unversity averages. (Two credit courses are multipled by two, etc.)
Center of Mass of an object is also commonly refered to as the Center of Gravity (CG) for the object.
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