Simple Harmonic Motion

9 26 pendulums we define the center of mass com of a

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Unformatted text preview: ums We define the center of mass (com) of a system of particles (such as a person) in order to predict the possible motion of the system. The center of mass of a system of particles is the point that moves as though (1) all of the system’s mass were concentrated there and (2) all external forces were applied there. In this section, we will discuss how to determine where the center of mass of a system of particles is located. We start with a system of only a few particles, and then we consider a system of a great many particles (a solid body, such as a baseball bat). 2.6.1 System of Particles Figure 9 shows two particles of mass m1 and m2 separated by distance d. We have arbitrary chosen the origin of an x axis to coincide with the particle of mass m1. We define the position of the center of mass (com) of this particle system to be (55) Now let us consider a more general situation, in which the coordinate system has been shifted leftward (See Figure 10). The position of the center of mass is no defined as (56) Note that if we put x1 = 0, then x2 becomes d and equation (56) becomes equation (55) as it must. Note also that in spite of the shift of the coordinate system, the center of mass is still the same distance from each particle. Figure 9 Center of mass of a system of two particles Now, if we extend equation (56) to a more general situation in which n particles are strung out along the x axis then the total mass becomes (57) in which M is the total mass of the system. 10 Thus we can write equation (57) as (58) 1 (59) The subscript i is a running number, or index that takes on all integer values from 1 to n. It identifies the various particles, their masses, and their x coordinates. If the particles are distributed in three dimensions, the center of mass must be identified by three coordinates. By extension of equation (59), the other coordinates are 1 1 1 (60) We can also define the center of mass with the language of vectors. First recall that the position of a particle at coordinates xi, yi, and zi is given by a position vector: ̂ (61) The position of the center of mass of the system of particles is given by a position vector: ̂ (62) Thus the three scalar equations of equations (60) can now be written as a single vector equation 1 (63) where again M is the total mass of the system. You can check that equation (63)...
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