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Unformatted text preview: Physics 235 Chapter 09- 1 - Chapter 9 Dynamics of a System of Particles In this Chapter we expand our discussion from the two-body systems discussed in Chapter 8 to systems that consist out of many particles. In general, these particles are exposed to both external and internal forces. In our discussion in the Chapter we will make the following assumptions about the internal forces: The forces exerted between any two particles are equal in magnitude and opposite in direction. The forces exerted between any two particles are directed parallel or anti-parallel to the line joining the two particles. These two requirements are fulfilled for many forces. However, there are important forces, such as the magnetic force, do not satisfy the second assumption. The Center-of-Mass As we discussed in Chapter 9, it is often useful to separate the motion of a system into the motion of its center of mass and the motion of its component relative to the center of mass. The definition of the position of the center of mass for a multi-particle system (see Figure 1) is similar to its definition for a two-body system: R cm = m i r i i m i i = 1 M m i r i i Figure 1. The location of the center of mass of a multi-particle system. If the mass distribution is a continuous distribution, the summation is replaced by an integration: Physics 235 Chapter 09- 2 - R cm = 1 M rdm Example: Problem 9.1 Find the center of mass of a hemispherical shell of constant density and inner radius r 1 and outer radius r 2 . Put the shell in the z &gt; 0 region, with the base in the x- y plane. By symmetry, x cm = y cm = To find the z coordinate of the center-of-mass we divide the shell into thin slices, parallel to the xy plane. z cm = zdV dV = zr 2 dr sin d d r = r 1 r 2 = 2 = 2 r 2 dr sin d d r = r 1 r 2 = 2 = 2 Using z = r cos and doing the integrals gives z cm = r 3 cos dr sin d d r = r 1 r 2 = 2 = 2 r 2 dr sin d d r = r 1 r 2 = 2 = 2 = 1 2 2 ( ) 1 4 r 2 4 r 1 4 ( ) 1 ( ) 2 ( ) 1 3 r 2 3 r 1 3 ( ) = 3 r 2 4 r 1 4 ( ) 8 r 2 3 r 1 3 ( ) Linear Momentum Consider a system of particles, of total mass M , exposed to internal and external forces. The linear momentum for this system is defined as P = m i r i i = d dt m i r i i = d dt M R ( ) = M R The change in the linear momentum of the system can be expressed in terms of the forces acting Physics 235 Chapter 09- 2 - R cm = 1 M rdm Example: Problem 9.1 Find the center of mass of a hemispherical shell of constant density and inner radius r 1 and outer radius r 2 ....
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- Fall '08