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| 183_notes:springmotion [2024/01/31 17:06] – caballero | 183_notes:springmotion [2026/01/04 20:19] (current) – hallstein |
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| {{url>https://glowscript.org/#/user/danny/folder/Shared/program/HorizontalSpring 700px,550px|Mass on a Spring}} | {{url>https://glowscript.org/#/user/danny/folder/Shared/program/HorizontalSpring 700px,525px|Mass on a Spring}} |
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| {{url>https://glowscript.org/#/user/danny/folder/Shared/program/SpringMassGraphs 700px,550px|Plot of spring-mass system}} | {{url>https://glowscript.org/#/user/danny/folder/Shared/program/SpringMassGraphs 700px,300px|Plot of spring-mass system}} |
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| You might have seen this kind of plot before. It's a [[http://en.wikipedia.org/wiki/Sine_wave|sinusoidal function]], in this case it's a sine curve. So the formula that describes this function could be something like: | You might have seen this kind of plot before. It's a [[http://en.wikipedia.org/wiki/Sine_wave|sinusoidal function]], in this case it's a sine curve. So the formula that describes this function could be something like: |
| For [[183_notes:constantf|constant force motion]], the time step ($\Delta t$) was not important because the average velocity ($\vec{v}_{avg} = \dfrac{\Delta \vec{r}}{\Delta t}$) and the //arithmetic// average velocity ($\vec{v}_{avg} = \dfrac{\vec{v}_f + \vec{v}_i}{2}$) were identical. When motion a system results from non-constant interactions, this is no longer true. Consider the figure below, which show predictions of the motion of a spring-mass system using different time steps. Here, you can clearly see that the smaller the time step, the more accurate the plot becomes (i.e., closer to the sinusoidal solution we expect). | For [[183_notes:constantf|constant force motion]], the time step ($\Delta t$) was not important because the average velocity ($\vec{v}_{avg} = \dfrac{\Delta \vec{r}}{\Delta t}$) and the //arithmetic// average velocity ($\vec{v}_{avg} = \dfrac{\vec{v}_f + \vec{v}_i}{2}$) were identical. When motion a system results from non-constant interactions, this is no longer true. Consider the figure below, which show predictions of the motion of a spring-mass system using different time steps. Here, you can clearly see that the smaller the time step, the more accurate the plot becomes (i.e., closer to the sinusoidal solution we expect). |
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| {{url>https://plot.ly/~dannycab/11/640/480 640px,480px|Plot of spring-mass system for different time steps}} | {{url>https://msuperl.org/interactive/mechanics/sinusoidal_position_vs_time_coarse_fine.html 640px,480px|Plot of spring-mass system for different time steps}} |
| ===== Examples ===== | ===== Examples ===== |
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| * [[183_notes:examples:Calculating the force due to a stretched spring]] | * [[183_notes:examples:Calculating the force due to a stretched spring]] |
| * [[183_notes:examples:Predicting the motion of system subject to a spring interaction]] | * [[183_notes:examples:Predicting the motion of system subject to a spring interaction]] |