Multi-particle Analysis of Momentum: Difference between revisions
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The Momentum Principle is <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings. | The Momentum Principle is <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings. | ||
The Energy Principle is <math>{Δ{E}} = {W}</math> where '''E''' is the total change of a system's energy ans '''W''' is the work done on the system by the surroundings. | |||
You will also need to know how to find the center of mass: <math>{cm} = {\frac{\sum{mr}}{{M}_{tot}}}</math>, where '''cm''' is the location of the center of mass relative to an origin, '''mr''' is a fractional mass and length product summed up to infinity, and '''M''' is the total mass of the system. | |||
===Procedure=== | |||
In order to analyze the motion of a multi-particle system, we need to apply both the energy principle and momentum principle. | |||
Step 0: Identify the system and the surroundings of the system. | |||
Step 1: Compress the system into a point-particle located at the system's center of mass. | |||
[insert image later] | |||
Step 2: Apply all forces acting on the system with their tails connected to the center of mass, retaining direction and magnitude. | |||
[insert image later] | |||
Step 3: Use the energy principle on the point particle. | |||
<math> Δ{K} = Δ{W}_{trans} = \vec{F} * Δ{r}_{cm}</math> | |||
Step 4: Return to the real-system and visualize the initial and final states of the systems. | |||
[insert images] | |||
Step 5: Calculate the work that each force does | |||
Step 6: Set up the Energy Principle for the problem. | |||
<math> Δ{K}_{trans} + Δ{K}_{rot} + Δ{U} = {W}_{surr} + {Q} </math> | |||
===A Computational Model=== | ===A Computational Model=== | ||
Revision as of 15:56, 5 December 2015
claimed by nacharya7
The Main Idea
The Momentum Principle is a foundation of classical physics that is applicable to almost any system of objects at any scale, micro- or macroscopic. As such it can be used to analyse systems of point particles as well as multi-particle systems; however, there are a few subtleties one needs to take note of when choosing to analyze a multi-particle system that this page will detail.
A Mathematical Model
The Momentum Principle is [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.
The Energy Principle is [math]\displaystyle{ {Δ{E}} = {W} }[/math] where E is the total change of a system's energy ans W is the work done on the system by the surroundings.
You will also need to know how to find the center of mass: [math]\displaystyle{ {cm} = {\frac{\sum{mr}}{{M}_{tot}}} }[/math], where cm is the location of the center of mass relative to an origin, mr is a fractional mass and length product summed up to infinity, and M is the total mass of the system.
Procedure
In order to analyze the motion of a multi-particle system, we need to apply both the energy principle and momentum principle.
Step 0: Identify the system and the surroundings of the system.
Step 1: Compress the system into a point-particle located at the system's center of mass.
[insert image later]
Step 2: Apply all forces acting on the system with their tails connected to the center of mass, retaining direction and magnitude.
[insert image later]
Step 3: Use the energy principle on the point particle.
[math]\displaystyle{ Δ{K} = Δ{W}_{trans} = \vec{F} * Δ{r}_{cm} }[/math]
Step 4: Return to the real-system and visualize the initial and final states of the systems.
[insert images]
Step 5: Calculate the work that each force does
Step 6: Set up the Energy Principle for the problem.
[math]\displaystyle{ Δ{K}_{trans} + Δ{K}_{rot} + Δ{U} = {W}_{surr} + {Q} }[/math]
A Computational Model
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