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183_notes:energy_sep [2014/10/31 15:20] – [(Near Earth) Gravitational Potential Energy] caballero | 183_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: |
+ | ==== Lecture Video ==== | ||
+ | {{youtube> | ||
==== 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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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.((Translation is motion that you have worked with in the past. It is the motion of point particles, no rotation or oscillation, | + | 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 |
- | + | ||
- | 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: | ||
==== Separating the Total Kinetic Energy in a Multi-Particle System ==== | ==== Separating the Total Kinetic Energy in a Multi-Particle System ==== | ||
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$$K_{tot} = K_{trans} + K_{rel}$$ | $$K_{tot} = K_{trans} + K_{rel}$$ | ||
- | This relative kinetic energy includes motion due to rotation about the center of mass (as in the above baton example) and oscillations or vibrations of the object. | + | This relative kinetic energy includes motion due to rotation about the center of mass (as in the above baton example) and oscillations or vibrations of the object. A derivation of this relationship (if you are interested) is [[: |
$$K_{rel} = K_{vib} + K_{rot}$$ | $$K_{rel} = K_{vib} + K_{rot}$$ | ||
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=== Translational Kinetic Energy === | === Translational Kinetic Energy === | ||
- | In physics, the word // | + | In physics, the word // |
$$K_{trans} = \dfrac{1}{2}M_{tot}v_{cm}^2$$ | $$K_{trans} = \dfrac{1}{2}M_{tot}v_{cm}^2$$ | ||
- | 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 ($v_{cm} | + | where the total mass of the system ($M_{tot}$) is under consideration. Here, you only consider systems moving slowly compared to the speed of light ($v_{cm} |
=== Vibrational Kinetic Energy === | === Vibrational Kinetic Energy === | ||
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$$K_{vib} = E_{tot} - (K_{trans} + U_s + E_{rest})\mathrm{\: | $$K_{vib} = E_{tot} - (K_{trans} + U_s + E_{rest})\mathrm{\: | ||
- | Here, the vibration energy can be calculated by knowing the other energy terms at a given time and location. A bit later in these notes, you will consider a spring-mass system that has all the different kinds of energy, and it will be a bit easier to see where the vibrational portion comes from. | + | 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 === | ||
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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), | ||
- | You will read much more about [[183_notes: | + | You will read much more about [[183_notes: |
==== (Near Earth) Gravitational Potential Energy ==== | ==== (Near Earth) Gravitational Potential Energy ==== | ||
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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 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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$$U_{tot} = \sum_i U_{atom,i} = \sum_i m_{atom, | $$U_{tot} = \sum_i U_{atom,i} = \sum_i m_{atom, | ||
- | This final sum is related to the [[183_notes: | + | This final sum is related to the [[183_notes: |
$$y_{cm} = \dfrac{1}{M_{tot}} \sum_i m_{atom, | $$y_{cm} = \dfrac{1}{M_{tot}} \sum_i m_{atom, | ||
$$M_{tot} y_{cm} = \sum_i m_{atom, | $$M_{tot} y_{cm} = \sum_i m_{atom, | ||
- | Hence, this sum can be replaced by the product of the total mass of the system and the location of the center | + | Hence, this sum can be replaced by the product of the total mass of the system and the location of the center |
- | $$U_{tot} = M_{tot}gy_{cm}$$. | + | $$U_{tot} = M_{tot}gy_{cm}$$ |
That is, the near Earth gravitational potential energy shared between a multi-particle system and the Earth, is mathematically equivalent to the energy shared by a point particle of mass $M_{tot}$ located at the center of mass, $y_{cm}$. | That is, the near Earth gravitational potential energy shared between a multi-particle system and the Earth, is mathematically equivalent to the energy shared by a point particle of mass $M_{tot}$ located at the center of mass, $y_{cm}$. |