Center of mass

The center of mass is the point where all of the mass of the object is concentrated. If the center of mass is a point within the object's actual structure, then the object can be balanced at that point. The object will also be free to rotate about that point.

FIGURE 5.3

To define the position of the center of mass for two particles, use the following formula:

For N particles the (vector) position of the CM is

r _CM =

Position of the center of mass(CM) r _CM =∑ m _i r _i (5.10)

where M = ∑ m _i, is the total mass of the system. The components of Eq. 5.10 are

x _CM = ; y _CM = ; z _CM = (5.11)

For a continuous distribution of mass, the expression for the center of mass of a collection of particles

x _CM = becomes an infinite sum and is expressed in the form of an integral

x _CM = (5.12)

The CM of a planar body can be determined experimentally as follows. When the body is pivoted freely about an arbitrary point, it behaves as if it were a simple pendulum with its mass

FIGURE 5.4

concentrated at the CM. The body will therefore rotate unless the CM lies vertically below the pivot, as shown in Fig. 5.4 a. In this position we draw a vertical line through the pivot. Next we suspend the body about another point, let it come to rest, and draw another vertical line. The intersection of the two lines, as indicated in Fig. 5.4 b, locates the CM. For a nonplanar body, one would use three points of suspension that do not lie in one plane.

Дата добавления: 2014-01-05