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183_notes:point_particle [2014/10/09 21:29] – [The Simplest System: A Single Particle] caballero | 183_notes:point_particle [2015/10/05 14:43] – [Work: Mechanical Energy Transfer] caballero | ||
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===== The Simplest System: A Single Particle ===== | ===== The Simplest System: A Single Particle ===== | ||
- | The [[183_notes: | + | The [[183_notes: |
==== Lecture Video ==== | ==== Lecture Video ==== | ||
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[{{183_notes: | [{{183_notes: | ||
- | The systems that you will consider will be approximated by a single object, the //point particle//. The point particle is an object that has no size of its own, but carries the mass of the object it is meant to represent. This point particle experiences the same force that the real object experiences, | + | The systems that you will consider will be approximated by a single object, the //point particle//. The point particle is an object that has no size of its own, but carries the mass of the object it is meant to represent. This point particle experiences the same force that the real object experiences, |
Thanks to Einstein, we know the total energy of a single particle system is given by, | Thanks to Einstein, we know the total energy of a single particle system is given by, | ||
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$$E_{tot} = \gamma m c^2$$ | $$E_{tot} = \gamma m c^2$$ | ||
- | where $m$ is the mass of the particle, $c$ is the speed of light in vacuum (3$\times$10^8 m/s), and $\gamma$ is the [[183_notes: | + | where $m$ is the mass of the particle, $c$ is the speed of light in vacuum (3$\times10^8$ m/s), and $\gamma$ is the [[183_notes: |
- | $$E_{tot} = \gamma m c^2 = \dfrac{1}{\sqrt{1-v^2/ | + | $$E_{tot} = \gamma m c^2 = \dfrac{1}{\sqrt{1-(v^2/c^2)}}mc^2 = \dfrac{1}{\sqrt{1-(0^2/c^2)}} mc^2 = mc^2$$ |
Evidently, a particle at rest has a total energy that is simply associated with its mass. This is called the // | Evidently, a particle at rest has a total energy that is simply associated with its mass. This is called the // | ||
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This is probably not the form of the kinetic energy that you are used to seeing. This is because for most purposes, objects are moving slowly enough where the relativistic correction doesn' | This is probably not the form of the kinetic energy that you are used to seeing. This is because for most purposes, objects are moving slowly enough where the relativistic correction doesn' | ||
- | $$K = (\gamma - 1)mc^2 = (\dfrac{1}{\sqrt{1-v^2/ | + | $$K = (\gamma - 1)mc^2 = \left(\dfrac{1}{\sqrt{1-v^2/ |
This definition of kinetic energy is due to Newton, but was confirmed by Coriolis and others. The total energy of a particle is thus the sum of its rest mass energy and its kinetic energy, which at low speeds is given by, | This definition of kinetic energy is due to Newton, but was confirmed by Coriolis and others. The total energy of a particle is thus the sum of its rest mass energy and its kinetic energy, which at low speeds is given by, | ||
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$$E_{tot} = E_{rest} + K = mc^2 + \dfrac{1}{2}mv^2$$ | $$E_{tot} = E_{rest} + K = mc^2 + \dfrac{1}{2}mv^2$$ | ||
- | For the time being you will neglect heat exchanges (although you will later relax that assumption), | + | For the time being you will neglect heat exchanges (although you will [[183_notes: |
$$\Delta E_{tot} = \Delta E_{rest} + \Delta K = W_{surr}$$ | $$\Delta E_{tot} = \Delta E_{rest} + \Delta K = W_{surr}$$ | ||
- | If the particle does not change | + | If the particle does not change |
$$\Delta K = K_f - K_i = W_{surr}$$ | $$\Delta K = K_f - K_i = W_{surr}$$ | ||
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$$K_f = K_i + W$$ | $$K_f = K_i + W$$ | ||
- | This is the update form of the [[183_notes: | + | This is the update form of the [[183_notes: |