183_notes:energy_sep

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183_notes:energy_sep [2014/11/05 17:39] – [The Total Kinetic Energy of a System is the Sum of All Its Parts] caballero183_notes:energy_sep [2018/05/29 21:50] – hallstein
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 +Section 9.1 in Matter and Interactions (4th edition) 
 +
 ===== Separating Energy in Multi-Particle Systems ===== ===== Separating Energy in Multi-Particle Systems =====
  
-You have read about the motion of the center of mass of a system from the perspective of the momentum principle. In these notes, you will read about how this motion can be connected to the energy of a multi-particle system, and how different kinetic energy terms can separated out from the total kinetic energy to be discussed and thought about separately.+You have read about the [[183_notes:center_of_mass|motion of the center of mass of a system]] from the perspective of the momentum principle. In these notes, you will read about how this motion can be connected to the energy of a multi-particle system, and how different kinetic energy terms can separated out from the total kinetic energy to be discussed and thought about separately. 
 +==== Lecture Video ====
  
 +{{youtube>Cobhu3lgeMg?large}}
 ==== The Total Kinetic Energy of a System is the Sum of All Its Parts ==== ==== The Total Kinetic Energy of a System is the Sum of All Its Parts ====
  
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 Now, consider that this baton is now tossed into the air while it twirls. The whole baton is moving up with a known speed. The kinetic energy of the baton has increased because the baton is both translating and rotating.((Translation is motion that you have worked with in the past. It is the motion of point particles, no rotation or oscillation, but it can be constant or accelerated motion.)) In this case, you can still add up the velocities of each atom, but now you have take into account the translational velocity of the whole baton.  Now, consider that this baton is now tossed into the air while it twirls. The whole baton is moving up with a known speed. The kinetic energy of the baton has increased because the baton is both translating and rotating.((Translation is motion that you have worked with in the past. It is the motion of point particles, no rotation or oscillation, but it can be constant or accelerated motion.)) In this case, you can still add up the velocities of each atom, but now you have take into account the translational velocity of the whole baton. 
  
-Consider a pair of atoms that are the same distance from the center of the baton (red circles in figure to the left). At this instant, the atom on the right is moving up as the baton rotates. The atom on the left is moving down. Relative to the fixed frame of the ground, the atom on the right, at this instant, is moving faster than the atom on the left. Adding up all the kinetic energies of the atoms here is a real pain. Luckily, we can separate the motion of the center of mass of the baton from the motion //around// the center of mass, making this energy calculation simpler. +Consider a pair of atoms that are the same distance from the center of the baton (red circles in figure to the right). At this instant, the atom on the right is moving up as the baton rotates. The atom on the left is moving down. Relative to the fixed frame of the ground, the atom on the right, at this instant, is moving faster than the atom on the left. This is another form of [[183_notes:relative_motion|the relative velocity motion that you read about earlier]]. Adding up all the kinetic energies of the atoms here is a real pain. Luckily, we can separate the motion of the center of mass of the baton from the motion //around// the center of mass, making this energy calculation simpler.
 ==== Separating the Total Kinetic Energy in a Multi-Particle System ====  ==== Separating the Total Kinetic Energy in a Multi-Particle System ==== 
  
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 $$U_{atom} = m_{atom}gy_{atom}$$ $$U_{atom} = m_{atom}gy_{atom}$$
  
-Those that are higher up will share more potential energy with the Earth than those lower to the ground. Those that are at the same height but different horizontal positions experience the same potential energy.((This argument requires all atoms to have the same mass, but van be extended to more general systems without loss of generality.))+Those that are higher up will share more potential energy with the Earth than those lower to the ground. Those that are at the same height but different horizontal positions experience the same potential energy.((This argument requires all atoms to have the same mass, but can be extended to more general systems without loss of generality.))
  
 If we consider a column of such atoms, that extends up some vertical height. The total potential energy associated with this column is given by the sum of the contributions due to each of the atoms, If we consider a column of such atoms, that extends up some vertical height. The total potential energy associated with this column is given by the sum of the contributions due to each of the atoms,
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