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183_notes:energy_sep [2021/04/17 17:26] – [Separating Energy in Multi-Particle Systems] stumptyl | 183_notes:energy_sep [2021/06/02 23:25] – [The Total Kinetic Energy of a System is the Sum of All Its Parts] stumptyl | ||
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- | ==== 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 ===== |
- | [{{ 183_notes:pp_vs_real.001.png? | + | [{{ 183_notes:week10_separation1.png? |
- | This might seem obvious to you, but you should realize that the total kinetic energy of any multi-particle system is the sum of all the individual kinetic energies of the particles or objects that make up the system. | + | This might seem obvious to you, but you should realize that the// total kinetic energy of any multi-particle system is the sum of all the individual kinetic energies of the particles or objects that make up the system//.// __The only caveat is that the velocity of all the constituent particles must be measured in the [[183_notes: |
$$K_{tot} = \sum_i K_i = \sum_i \dfrac{1}{2}m_iv_i^2$$ | $$K_{tot} = \sum_i K_i = \sum_i \dfrac{1}{2}m_iv_i^2$$ | ||
- | === Twirling a Baton === | + | ==== Twirling a Baton ==== |
[{{ 183_notes: | [{{ 183_notes: | ||
- | Consider a baton that is being twirled in a circle in someone' | + | Consider a baton that is being twirled in a circle in someone' |
- | 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 | + | 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 the motion that you have worked |
- | 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: | + | Consider a pair of atoms that are the same distance from the center of the baton (red circles in the 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: |
- | ==== Separating the Total Kinetic Energy in a Multi-Particle System ==== | + | ===== Separating the Total Kinetic Energy in a Multi-Particle System |
The total kinetic energy of a multi-particle system easily separates into the translational kinetic energy associated with the motion of the center of mass and the motion relative to the center of mass. | The total kinetic energy of a multi-particle system easily separates into the translational kinetic energy associated with the motion of the center of mass and the motion relative to the center of mass. | ||
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There is a formal derivation of this equality, but it is more important that you understand conceptually the idea of separating the energy due to translation of the center of mass ($K_{trans}$) and energy due to motion relative to the center of mass ($K_{vib}$ and $K_{rot}$). | There is a formal derivation of this equality, but it is more important that you understand conceptually the idea of separating the energy due to translation of the center of mass ($K_{trans}$) and energy due to motion relative to the center of mass ($K_{vib}$ and $K_{rot}$). | ||
- | === Translational Kinetic Energy === | + | ==== Translational Kinetic Energy |
- | In physics, the word //translation// means to move from one location to a different location. When you are interested in how a multi-particle system moves, you might want to track how the whole system moves from one location to another. This motion is captured by the motion of the center of mass. That is, you are avoiding the details of how the system rotates or vibrates and are just following the " | + | In physics, the word **translation** means to move from one location to a different location. When you are interested in how a multi-particle system moves, you might want to track how the whole system moves from one location to another. This motion is captured by the motion of the center of mass. That is, you are avoiding the details of how the system rotates or vibrates and are just following the " |
$$K_{trans} = \dfrac{1}{2}M_{tot}v_{cm}^2$$ | $$K_{trans} = \dfrac{1}{2}M_{tot}v_{cm}^2$$ | ||
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where the total mass of the system ($M_{tot}$) is under consideration. Here, you only consider systems moving slowly compared to the speed of light ($v_{cm} \ll c$). | where the total mass of the system ($M_{tot}$) is under consideration. Here, you only consider systems moving slowly compared to the speed of light ($v_{cm} \ll c$). | ||
- | === Vibrational Kinetic Energy === | + | ==== Vibrational Kinetic Energy |
You read about the [[183_notes: | You read about the [[183_notes: | ||
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Here, the vibration energy can be calculated by knowing the other energy terms at a given time and location. | Here, the vibration energy can be calculated by knowing the other energy terms at a given time and location. | ||
- | === Rotational Kinetic Energy === | + | ==== Rotational Kinetic Energy |
Just as there can be kinetic energy associated with vibrations without motion of the center of mass (i.e., no translation), | Just as there can be kinetic energy associated with vibrations without motion of the center of mass (i.e., no translation), |