sys - Physics 53 Systems of Particles Everything should be...

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Physics 53 Systems of Particles Everything should be as simple as it is, but not simpler. — Albert Einstein Overview An object of ordinary size — which we call a “macroscopic” system — contains a huge number of atoms or molecules. It is out of the question to attempt to use the laws we have discussed for a single particle to describe separately each particle in such a system. There are nevertheless some relatively simple aspects of the behavior of a macroscopic system that (as we will show) follow from the basic laws for a particle. We will be able to show that in any multi-particle system each of the mechanical quantities relevant to the state of the system consists of two parts: 1. One part in which the system is treated as though it were a single particle (with the total mass of the system) located at a special point called the center of mass . This part is often called the CM motion . 2. Another part describing the internal motion of the system, as seen by an observer located at (and moving with) the center of mass. In this section we will give a general analysis of the variables to be used in describing the motion of any system of particles, large or small. In later sections we will apply this analysis to the case of solid objects, in the approximation that all the particles in the object have a Fxed spatial relation to each other. In this approximation, the object is called a “rigid body.” Still later we will discuss ±uids, including gases, where things are more complicated because the internal motion is dominant, and where we must take a statistical approach in the analysis. Center of mass We consider a system composed of N point particles, each labeled by a value of the index i which runs from 1 to N . Each particle has its own mass m i and (at a particular time) is located at its particular place r i . The center of mass (CM) of the system is deFned by the following position vector: PHY 53 1 Systems
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Center of mass r CM = 1 M m i r i i where M is the total mass of all the particles. This vector locates a point in space — which may or may not be the position of any of the particles. It is the mass-weighted average position of the particles, being nearer to the more massive particles. As a simple example, consider a system of only two particles, of masses m and 2 m , separated by a distance . Choose the coordinate system so that the less massive particle is at the origin and the other is at x = , as shown in the drawing. Then we have m 1 = m , m 2 = 2 m , x 1 = 0 , x 2 = . (The y and z coordinates are zero of course.) We Fnd from the deFnition x CM = 1 3 m m 0 + 2 m ( ) = 2 3 . The location is indicated on the drawing. As time goes on the position vectors of the particles
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sys - Physics 53 Systems of Particles Everything should be...

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