Iterative Prediction of Spring-Mass System: Difference between revisions

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The Momentum Principle provides a mathematical basis for the repeated calculations needed to predicts the system's future motion.
The Momentum Principle provides a mathematical basis for the repeated calculations needed to predicts the system's future motion.


'''Momentum Principle''': <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math>
The most useful form of this equation is referred to as the ''momentum update form'', and can be derived by rearranging the Momentum Principle as shown below:


The most useful form of this equation is referred to as the "momentum update form", and can be derived by rearranging the Momentum Principle as shown below:
<div style="text-align: center;"><math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math>


<div style="text-align: center;"><math>{\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{&Delta;t}}</math>
<math>{\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{&Delta;t}}</math>


<math>{\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{&Delta;t}}</math>
<math>{\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{&Delta;t}}</math>
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<div style="text-align: left;">
<div style="text-align: left;">
Together, these equations can be used to model the future motion of a spring-mass system.




Revision as of 15:03, 1 December 2015

claimed by kgiles7

Short Description of Topic

The Main Idea

A simple spring-mass system is a basic illustration of the momentum principle. The principle of conservation of momentum can be repeatedly applied to predict the system's future motion.


A Mathematical Model

The Momentum Principle provides a mathematical basis for the repeated calculations needed to predicts the system's future motion.

The most useful form of this equation is referred to as the momentum update form, and can be derived by rearranging the Momentum Principle as shown below:

[math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math]

[math]\displaystyle{ {\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{&Delta;t}} }[/math]

[math]\displaystyle{ {\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{&Delta;t}} }[/math]


In order to update the object's velocity and position, similar equations can be used:

Velocity Update Formula: [math]\displaystyle{ {\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{&Delta;t} }[/math]

Position Update Formula: [math]\displaystyle{ {\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{&Delta;t}} }[/math]

Together, these equations can be used to model the future motion of a spring-mass system.


A Computational Model

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