Multi-particle Analysis of Momentum

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.

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Step 2: Apply all forces acting on the system with their tails connected to the center of mass, retaining direction and magnitude.

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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.

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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

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Problem Description

Two wooden blocks of mass [math]\displaystyle{ {\vec{m}} }[/math] are shot by a bullet on a vertical apparatus such that the first wooden block along the axis of its center of mass while the other is hit slightly to the left. The bullet travels at speed [math]\displaystyle{ {\vec{v}} }[/math]. Which block travels further?

Solution

Step 0: System: Block, Bullet Surroundings: Earth

Step 1:

After compressing each block to their centers of masses, ...

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