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Demonstration for a spring and mass systems on a surface without friction

Posted 1 year ago
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This idea came from my need to demonstrate a simple physical phenomenon such as the motion of a spring mass system to my undergraduate students. In the absence of a live demo, built-in Wolfram Language functions such as Animate and Manipulate come in handy. Here I will try to show a few simple examples.

In order to set up a stationary spring-mass system where the spring is tied to a fixed wall on one side and a mobile block on the other end, I used a simple graphics setup as follows:

Graphics[{Thickness[0.01], Line[{{0, 1}, {0, 0}, {4, 0}}], {Dashed, Line[{{x, 1}, {x, 0}, {x, -0.5}}], 
            Line[{{2, 1}, {2, 0}, {2, -0.5}}], Line[{{0, 0}, {0, -0.5}}]}, {EdgeForm[Thick], Pink, Opacity[0.5], 
                                                                              Rectangle[{x + .01, 0.01}, {x + 0.76, 0.76}]}}]

Block on surface

ListLinePlot[Table[{(x - 0.19)/3900 i + 0.1 (1 - Cos[\[Pi]/100 (i - 1)]), 0.36 + 0.1 Sin[\[Pi]/100 (i - 1)]}, {i, 1, 3901}], 
                               AspectRatio -> 1, PlotStyle -> Black, Axes -> False]

Spring image

Show[Graphics[{Thickness[0.01], Line[{{0, 1}, {0, 0}, {4, 0}}], 
             {Arrowheads[0.03], Thickness[0.005], Arrow[{{0.8, -0.15}, {0, -0.15}}], Arrow[{{1, -0.15}, {2, -0.15}}], 
                Text[Style["\!\(\*SubscriptBox[\(x\), \(0\)]\)", FontSize -> 30], {0.9, -0.15}], 
                    If[x > 2.3 \[Or] x < 1.7, {Arrowheads[{-0.03, 0.03}], Arrow[{{2, -0.3}, {x, -0.3}}], 
                                                               Text[Style["x", FontSize -> 30], {0.5 x + 1, -0.45}]}], 
             {Red, Arrow[{{x, 0.85}, {2, 0.85}}], Text[Style["F = - k x", FontSize -> 30], {2, 1.1}]}, 
             {Dashed, Line[{{x, 1}, {x, 0}, {x, -0.5}}], Line[{{2, 1}, {2, 0}, {2, -0.5}}], Line[{{0, 0}, {0, -0.5}}]}}, 
             {EdgeForm[Thick], Pink, Opacity[0.5], Rectangle[{x + .01, 0.01}, {x + 0.76, 0.76}]}}],
         ListLinePlot[Table[{(x - 0.19)/3900 i + 0.1 (1 - Cos[\[Pi]/100 (i - 1)]), 0.36 + 0.1 Sin[\[Pi]/100 (i - 1)]}, 
                              {i, 1, 3901}], 
         AspectRatio -> 1, PlotStyle -> Black, Axes -> False], PlotRange -> {{0, 4}, {-0.7, 1.5}}]

Spring & mass

In the above figure I have also added more information using Text[] ...

Now we are at a position to animate the above figure and see if what we get.

Animate[Show[Graphics[{Thickness[0.01], Line[{{0, 1}, {0, 0}, {4, 0}}], 
                 {Arrowheads[0.03], Thickness[0.005], Arrow[{{0.8, -0.15}, {0, -0.15}}], Arrow[{{1, -0.15}, {2, -0.15}}], 
                     Text[Style["\!\(\*SubscriptBox[\(x\), \(0\)]\)", FontSize -> 30], {0.9, -0.15}], 
                      If[x > 2.3 \[Or] x < 1.7, {Arrowheads[{-0.03, 0.03}], Arrow[{{2, -0.3}, {x, -0.3}}], 
       Text[Style["x", FontSize -> 30], {0.5 x + 1, -0.45}]}], {Red, Arrow[{{x, 0.85}, {2, 0.85}}], 
      Text[Style["F = - k x", FontSize -> 30], {2, 1.1}]},
                    {Dashed, Line[{{x, 1}, {x, 0}, {x, -0.5}}], Line[{{2, 1}, {2, 0}, {2, -0.5}}], Line[{{0, 0}, {0, -0.5}}]}}, 
      {EdgeForm[Thick], Pink, Opacity[0.5], Rectangle[{x + .01, 0.01}, {x + 0.76, 0.76}]}}], 
             ListLinePlot[Table[{(x - 0.19)/3900 i + 0.1 (1 - Cos[\[Pi]/100 (i - 1)]),  0.36 + 0.1 Sin[\[Pi]/100 (i - 1)]}, 
                                      {i, 1, 3901}], AspectRatio -> 1, PlotStyle -> Black, Axes -> False], 
                     PlotRange -> {{0, 4}, {-0.7, 1.5}}], {x, 1, 3}, AnimationRunning -> False, AnimationRepetitions -> 2,                            AnimationDirection -> ForwardBackward]

Full motion gif

Animation

The result for this reasonably represents the motion of the block on the smooth floor (with scope of improvement :))

2 Replies

Thanks for sharing! I have not seen exactly this setup at Wolfram Demonstrations. Did you try to search for a similar system there for the class you teach? If your example is uniques you should submit this to Wolfram Demonstrations, so other educators would be able to get use of it.

http://demonstrations.wolfram.com/search.html?query=spring%20mass

enter image description here

Thanks for your comment Vitaliy. I have seen some of the demonstrations. They are quite good and in some occasions too good. My post was to show something at a much lower level in the form of an interactive picture for demonstrating for instance the direction of force etc. :)

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