simple harmonic motion

Spring Motion and Numerical Calculations

rallain — Thu, 16 Oct 2008 20:57:42 +0000

Spring Motion and Numerical Calculations

Maybe you know I like numerical calculations, well I do. I think they are swell. [VPython](http://vpython.org) is my tool of choice. In the post [Basics: Numerical Calculations](http://scienceblogs.com/dotphysics/2008/10/basics-numerical-calculation…) I used vpython and excel to do something simple. I will do that again today (in that this problem could also be solved analytically). However, there is one big difference. This problem has a non-constant forces. Suppose I have a mass that is connected by a spring to a wall. This mass-spring is sitting on a table with no friction.

![Screenshot 27](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

There is a very interesting property of springs. The more you stretch them, the greater the force they exert (in the usual model of springs). This model works very well.

![Screenshot 28](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

This is known as Hooke's law. I have written it as a scalar for simplicity. The "k" is called the spring constant. It is a measure of how "stiff" the spring is. The value "s" is the amount the spring is stretched. Typically, there is a minus sign in front of the ks to indicate that the force is in the opposite direction that the spring is stretched. Really, in a scalar equation this is rather silly to include (but everyone does anyway).

**Question: What will the motion of the mass be like if I pull it back and then let go?**

Although this can be determined analytically, I am going to first calculate this with vpython. I will try to show all the details so that you can reproduce this also. If you have not already installed [vpython](http://vpython.org), do that now (don't cost nothing).

In the IDLE editor, enter the following:

![Screenshot 30](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

The first line (it's two underscores, then future, then two more underscores) imports a better division than the standard python division.
The next line imports all the vpython stuff. This lets you use functions like "sphere(), and vector()".
The last import makes it so it is easy to create a graph.

Now you should save and run your program just to make sure everything is ok. If it is fine, you will get a yellow sphere.

Ok, I trust it worked. Here is the plan for the program.

Set up constants and set up stuff for the graph
Make a loop
In the loop, update the force (force of the spring depends on position)
Use the force to update the momentum
Update the position using the change in time and the momentum divided by the mass
Update the graph
Do the loop again....and again....

Note that I am only representing the ball, not the floor or wall or spring. Also, I am going to set up a weird spring. This spring is attached at the origin and as a zero natural length. This means that any displacement of the mass from the origin will result in a force proportional to that displacement. This is not realistic, but easy to calculate and it gets the point across.

So, let me get started with the setup part:

![Screenshot 31](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

I calculated the initial momentum as the mass times the vector(0,0,0) which is STILL the zero vector. However, this way I could easily go back and put the initial velocity as something else.
ks is the spring constant
I set the time interval to be small. If you have this too large, things don't work too well. It IS possible to fix this by changing the "recipe" a little, but I want it to look simple.
posgraph sets up the graph that I will make

If you want, you can run the program (it won't do anything) to see if you have any syntax errors.

It works? Ok, here is the rest of the program.

![Screenshot 32](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

First thing, I changed the position of the mass to pos=(.5, 0, 0). If I didn't do this, it would never move because there would be no force to cause a change in momentum. Of course, it would work if I left it at the origin and gave it an initial momentum - try that.
The fnet vector is just the opposite of the position from the origin of the mass (times some constant). If you want, you can make your mass go in any direction
posgraph.plot puts a point on the graph. It plots the point at pos=(horizontal value, vertical value). In this case, I want time on the horizontal axis and just the x-component of the position of the mass.

Now run your program. It will happen fast because computers are fast (there is a way to slow down the motion, but I am too impatient for that). Hopefully, your program will produce a graph that looks like this:

![Screenshot 34](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

This is probably more awesome than you realize. What function does this graph look like? If you had to guess? Does it kind of look like the cosine function? Why is that awesome?

**What is cosine**

The cosine function is simply the ratio of the sides a and c in the diagram below of a right triangle.

![Screenshot 35](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

Cosine is can also be explained as the projection of a radius of a circle onto one of the axes.

![Screenshot 36](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

As the red line moves around the circle, the length of the green line changes. The ratio of the green line to the red line is the cosine of the angle of the red line. So, the sine and cosine functions are associated with triangles and circle (and ? - sorry, I just wanted to give a shout out to ?). Well, who cares. Obviously I care. I care because here in the program, do you see a circle? No. Do you see a triangle? No. Do you see ?? No. But, the program clearly produces a cosine function. I think that is awesome.

Well, maybe the program is wrong. No, it's not wrong. You can easily set up an experiment and get similar data. Also, I can do this problem analytically and get the same answer.

**Analytical Solution**

For this solution, I will already assume that all motion takes place in the x-direction. So, I can write Newton's second law as:

![Screenshot 37](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

Where F_x is the total force (there is only one in this case anyway). Please forgive me, but I am going to drop the "in the x-direction" notation since EVERYTHING is only in the x-direction. The force from the spring is:

![Screenshot 38](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

Remember that this spring has zero natural length, so the x position IS the "stretch". Putting this together, I get:

![Screenshot 39](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

Here I wrote acceleration as the second time derivative of position. If that is something completely foreign to you, don't worry - you will see this later maybe. Anyway, what I have here is a differential equation. How do you solve a differential equation? There are lots of strategies, but I find the best one is to guess. Yes, just guess a solution and see if it works. First, let me re-write the differential equation:

![Screenshot 40](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

If you look at this equation, it says "take the derivative with respect to time twice and get something times the original function" (really, it says that, you might have to listen closely). Once function that does that is ....cosine. So, let me try the function:

![Screenshot 41](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

Where A and ? are constants. Let me take the first derivative:

![Screenshot 42](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

And now the second derivative:

![Screenshot 43](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

So this means that:

![Screenshot 44](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

Well, does this agree with the numerical solution? One easy way to compare is the period. The period is:

![Screenshot 46](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

Using the values from the original situation:

![Screenshot 47](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

Is this what the numerical calculation gives also? I will add the following inside the while loop:

![Screenshot 48](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

This should print the time if the position of the mass is near where it started. The output is:

![Screenshot 49](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

So from this, the period is right around 1.256 seconds. They agree. I think this is an excellent example of how numerical calculations are really the same thing as analytical calculations. Ok, they are not the same, but they do the same thing.

rallain Thu, 10/16/2008 - 16:57

Resonance and a Magic Trick

rallain — Sat, 04 Oct 2008 20:12:51 +0000

Resonance and a Magic Trick

Magic tricks are cool. Especially when the trick is really physics. In this trick, I can make one of the four balls move more than the others. (When you watch the video, you will see why I am not a professional magician). You could set this up in a variety of ways. I state that if we (me and people around me) all work together with our mind and focus on the same ball, our brain waves can resonate with that ball and make it move. I let the people around me pick. In this video, I make the smallest two move.

So, what is the trick? The trick is not a trick. It is not resonance with brain waves, but it is resonance.

Each one of those balls, if displaced, will oscillate at a particular frequency. For a pendulum, this frequency is:

![Screenshot 22](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

Where *g* is the local gravitational field (9.8 N/kg or 9.8 m/s²) and *l* is the length of the pendulum. So, each ball has a different length and thus a different frequency that it will swing back and forth at. If you drive (shake) that pendulum at the frequency that it naturally swings, the amplitude of its oscillation will get bigger. You have already seen this effect and used it. I know you have. If you have ever pushed a child (or adult) on a swing, you know that you can't just push them when ever you want. If you wait and push while they are the top of their swing, you will increase the amplitude of the motion. This is resonance.

So, in the magic trick, you just need to slightly shake the stick (like I said, I am not very good so you can probably notice that I am shaking it). If you shake at the frequency for one of the balls, its amplitude will increase. Well, how do you know what frequency? Do you have to calculate it before hand? No, just focus on the one you want to move and shake in sync with its motion.

To take this idea a little further than it needs to go (isn't that what I always do?), I decided to model this situation. Modeling a pendulum with a moving point isn't nearly as easy as an oscillating spring with an oscillating point. Both can be used to demonstrate resonance. In my model (created in [vpython](http://www.vpython.org)), I have 4 masses attached by springs to a moving wall. The four mass all have different masses. For a mass on a spring, the frequency of oscillation is:

![Screenshot 23](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

In the case of the 4 masses with different length strings, the length changes and *g* does not. To make the situation similar, I will keep the spring constant (*k*) constant and change the mass.

(Hopefully I will remember to talk about the oscillation of a spring - it is awesome on so many different levels)

In this model, I have 4 different masses and each one has a particular "natural" frequency. If I shake the wall at the natural frequency of one of these masses (even if I shake it a little) the wall keeps pushing the mass at the right time making the amplitude larger. For the other masses, the pushes are not at the right time and they do not get bigger. Here is the position of the 4 masses when the wall oscillates at the frequency for mass 1:

![Screenshot 24](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

The green line represents the position of mass 1. Note the black line is the position of the wall. It has a very small amplitude, but still produces significant motion in mass 1. The other masses still move, but they do not significantly increase in amplitude. What if I oscillate the wall at the frequency for mass 2?

![Screenshot 25](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

And here are the graphs for resonance at the frequency for mass 3 and 4:

![Screenshot 26](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screensho…)

**The Trick**

So the trick is to move that stick at a frequency that is the same as one of the oscillators. If you move it just a little (small amplitude) maybe people won't notice and think you are Harry Potter or something.

rallain Sat, 10/04/2008 - 16:12