So far we have looked at systems of point masses on a line and in a plane. The center of mass, is the point where the fulcrum should be placed to make the system balance. Suppose we have two point masses, and located on a number line at points and respectively ( (Figure)). However, we are really interested in systems in which the masses are not allowed to move, and instead we balance the system by moving the fulcrum.
In the seesaw example, we balanced the system by moving the masses (children) with respect to the fulcrum. Applying this concept to the masses on the rod, we note that the masses balance each other if and only if
If the heavier child slides in toward the center, though, the seesaw balances. On a seesaw, if one child sits at each end, the heavier child sinks down and the lighter child is lifted into the air. The most common real-life example of a system like this is a playground seesaw, or teeter-totter, with children of different weights sitting at different distances from the center. Last, we use centroids to find the volume of certain solids by applying the theorem of Pappus.įigure 1. In this section, we first examine these concepts in a one-dimensional context, then expand our development to consider centers of mass of two-dimensional regions and symmetry. (That is why performers spin the plates the spin helps keep the plates from falling even if the stick is not exactly in the right place.) Mathematically, that sweet spot is called the center of mass of the plate. If we put the stick anywhere other than that sweet spot, the plate does not balance and it falls to the ground. If we look at a single plate (without spinning it), there is a sweet spot on the plate where it balances perfectly on the stick. The performers try to keep several of them spinning without allowing any of them to drop. Many of us have seen performers who spin plates on the ends of sticks. The basic idea of the center of mass is the notion of a balancing point. In this section, we consider centers of mass (also called centroids, under certain conditions) and moments. Apply the theorem of Pappus for volume.Use symmetry to help locate the centroid of a thin plate.Locate the center of mass of a thin plate.
Find the center of mass of objects distributed along a line.
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