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--- 183_notes:point_particle [2014/10/03 21:08] – caballero
+++ 183_notes:point_particle [2021/05/06 20:42] (current) – [The Total Energy of a Single Particle] stumptyl
@@ Line 1: / Line 1: @@
+Section 6.2 in Matter and Interactions (4th edition)
 ===== The Simplest System: A Single Particle =====
-The [[183_notes:define_energy#the_first_law_of_thermodynamics_the_energy_principle|energy principle]] is widely applicable and helps to explain or to predict the motion of systems by considering how the system exchanges energy with its surroundings. For now, you will read about the simplest of systems that of a single particle. Here, you will read about the total energy of a particle, the energy due to its motion, and how those energies are connected in situations where we can neglect the heat exchanges.
+The [[183_notes:define_energy#the_first_law_of_thermodynamics_the_energy_principle|energy principle]] is widely applicable and helps to explain or to predict the motion of systems by considering how the system exchanges energy with its surroundings. For now, you will read about the simplest of systems, that of a single particle. **In these notes, you will read about the total energy of a particle, the energy due to its motion, and how those energies are connected in situations where we can neglect the [[183_notes:heat|heat exchanges]]**.
+==== Lecture Video ====
+{{youtube>jZ6aQdLdRUM?large}}
 ==== The Total Energy of a Single Particle ====
-[{{183_notes:real_to_pp.001.png?500|A real car crushed down to a point particle for the purpose of modeling the translation of the car. }}]
+[{{  183_notes:week7_cartopoint.png?500|A real car crushed down to a point particle for the purpose of modeling the translation of the car. }}]
-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, and thus models the motion of that real physical system to the extent that you only care about how the object translates (moves without rotation). Point particles do not spin or change their shape. Later, we will relax these conditions.
+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, and thus models the motion of that real physical system to the extent that you only care about how the object translates (moves without rotation). Point particles do not spin or change their shape. Later, [[183_notes:energy_sep|we will relax these conditions]].
 Thanks to Einstein, we know the total energy of a single particle system is given by,
@@ Line 12: / Line 17: @@
  $E_{tot} = \gamma m c^2$
-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:momentum#when_does_the_gamma_factor_matter|correction due to relativity]] when the particle is moving near the speed of light. If a system of one particle is at rest ( $v=0$ ) then,
+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:momentum#when_does_the_gamma_factor_matter|correction due to relativity]] when the particle is moving near the speed of light. If a system of one particle is at rest ( $v=0$ ) then,
- $E_{tot} = \gamma m c^2 = \dfrac{1}{\sqrt{1-v^2/c^2}}mc^2 = \dfrac{1}{\sqrt{1-0^2/c^2}} mc^2 = mc^2$
+$$E_{tot} = \gamma m c^2 = \dfrac{1}{\sqrt{1-(v^2/c^2)}}mc^2 = \dfrac{1}{\sqrt{1-(0^2/c^2)}} mc^2 = mc^2$$
 Evidently, a particle at rest has a total energy that is simply associated with its mass. This is called the //[[http://en.wikipedia.org/wiki/Mass–energy_equivalence|rest mass energy]]// of that particle and really matters when particles change their identity (e.g., in chemical or nuclear reactions).
@@ Line 20: / Line 25: @@
  $E_{rest} = mc^2$
-It appears that the rest of the energy is associated with the motion of the particle. As such, it is refereed to as the //kinetic energy// of the particle.
+It appears that the rest of the energy is associated with the motion of the particle. As such, it is refereed to as the __**kinetic energy (J)**__ of the particle.
  $K = E_{tot} - E_{rest} = \gamma m c^2 - mc^2 = (\gamma - 1)mc^2$
@@ Line 26: / Line 31: @@
 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't matter. At low speeds,
- $K = (\gamma - 1)mc^2 = (\dfrac{1}{\sqrt{1-v^2/c^2}}-1) mc^2 \approx \left(\left(1+\dfrac{1}{2}\dfrac{v^2}{c^2}\right)-1\right)mc^2 = \dfrac{1}{2}\dfrac{v^2}{c^2} mc^2 = \dfrac{1}{2}mv^2$
+$$K = (\gamma - 1)mc^2 = \left(\dfrac{1}{\sqrt{1-v^2/c^2}}-1\right) mc^2 \approx \left(\left(1+\dfrac{1}{2}\dfrac{v^2}{c^2}\right)-1\right)mc^2 = \dfrac{1}{2}\dfrac{v^2}{c^2} mc^2 = \dfrac{1}{2}mv^2$$
 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,
@@ Line 32: / Line 37: @@
  $E_{tot} = E_{rest} + K = mc^2 + \dfrac{1}{2}mv^2$
-For the time being you will neglect heat exchanges (although you will later relax that assumption), so that the system of a single particle system changes it's total energy as a result of work by the surroundings.
+For the time being you will neglect heat exchanges (although you will [[183_notes:heat|later relax that assumption]]), so that the system of a single particle system changes its total energy as a result of work by the surroundings.
  $\Delta E_{tot} = \Delta E_{rest} + \Delta K = W_{surr}$
-If the particle does not change it's identity, then there's no change in rest mass energy and you are left with,
+If the particle does not change its identity, then there is no change in rest mass energy and you are left with,
  $\Delta K = K_f - K_i = W_{surr}$
@@ Line 69: / Line 74: @@
  $K_f = K_i + W$
-This is the update form of the [[183_notes:define_energy#the_first_law_of_thermodynamics_the_energy_principle|energy principle]] for a single particle that doesn't change it's identity.
+This is the update form of the [[183_notes:define_energy#the_first_law_of_thermodynamics_the_energy_principle|energy principle]] for a single particle that doesn't change its identity.