What could be causing the high energy losses in the pendulum?

[Just a nobody]

In my intro mechanics lab class, we did an experiment where we measured how much energy was lost in a pendulum over 300 seconds. In my case, the final energy of the pendulum was 2.4% what the initial energy was.

The pendulum was constructed by attaching a cylindrical metal bob to a thread.

Today I tried to simulate that system using a computer. I came up with this equation of motion:

$\ddot{\theta} = - \frac{g}{l} \sin \theta - \frac{1}{2m} C_D \rho A l \frac{\dot{\theta}^3}{|\dot{\theta}|}$

Where the last term is due to the air resistance on the bob:

$F_D = \frac{1}{2} C_D \rho A v^2$

I estimated the drag coefficient $C_D$ to be somewhere between 0.5 and 1, based on the values I saw in the figure in the top right of http://en.wikipedia.org/wiki/Drag_coefficient" .

When I ran the simulation, the final energy/initial energy ratio turned out to be 33%, even taking $C_D$ to be my maximum estimated value, 1. So what I'm wondering is: what other sources of energy loss might there be that I'm not thinking of? I realize that I'll never be able to get the exact value, but I should be able to get something closer to 2.4% than 33% is.

I attached the program (it's in C) in case anyone cares to take a look. If you have gcc, you can compile with `gcc -o pendulum -lm pendulum.c` and run with `./pendulum`.

 

[Dadface]

Hello just a nobody.The vibrating pendulum sets the support structure into vibration as well.Depending on how well this is constructed and set up it can be a major source of energy loss.

 

[phyzguy]

Interesting. It looks to me like you've done it all correctly. The problem is probably that there are other energy loss mechanisms, as you say. In addition to what Dadface says, there might be drag on the string itself that you could add in, and there might be energy losses from flexing the string back and forth. You might try fastening it to a more rigid support to eliminate the losses Dadface refers to, and using some sort of bearing at the pivot point to minimize those losses. Another thing you could try is to use multiple bobs of different areas and plot the energy loss as a function of area, then extrapolate back to zero area. Then you could quantify how much of the energy loss comes from drag on the bob and how much comes from "other stuff". Then you could work to minimize the "other stuff"

 

[Redbelly98]

EDIT: what I say in this post is, for the most part, not useful.

So what I'm wondering is: what other sources of energy loss might there be that I'm not thinking of? I realize that I'll never be able to get the exact value, but I should be able to get something closer to 2.4% than 33% is.

[STRIKE]I think it's impossible that is the problem, see my note below.[/STRIKE]

[STRIKE]Your parameters all look reasonable, so I wonder about some bug in the code. Can you confirm that the period is what it should be, based on the value of g and the pendulum length? Also, did the experiment and simulation use the same starting angle? The percentage loss would be greater for larger angles, owing to the larger velocity and v² dependence of the drag force.[/STRIKE]

Hello just a nobody.The vibrating pendulum sets the support structure into vibration as well.Depending on how well this is constructed and set up it can be a major source of energy loss.

Interesting. It looks to me like you've done it all correctly. The problem is probably that there are other energy loss mechanisms, as you say. In addition to what Dadface says, there might be drag on the string itself that you could add in, and there might be energy losses from flexing the string back and forth. You might try fastening it to a more rigid support to eliminate the losses Dadface refers to, and using some sort of bearing at the pivot point to minimize those losses.

(EDIT: the following paragraph is based on my misunderstanding.)

[STRIKE]I have to disagree with the way the three of you are thinking about other losses to explain the results. The simulation shows more loss than the actual experiment. Trying to come up with other sources of loss, and including them in the simulation, will only worsen the discrepancy. Likewise, eliminating those other losses from the experiment would worsen the discrepancy as well.

I do find myself second guessing what I am saying in the preceding paragraph, based on the fact that I am outvoted 3 to 1! Do I misunderstand something?[/STRIKE]

I have some Runge Kutta code as well, and will take a look at using justanobody's parameters and see what I get.

 

[Dadface]

I think there is a misunderstanding here.As I read it the loss was not 2.4 percent because in the experiment the final energy was 2.4 percent of the original meaning a loss of 97.6 percent.

 

[Redbelly98]

I think there is a misunderstanding here.As I read it the loss was not 2.4 percent because in the experiment the final energy was 2.4 percent of the original meaning a loss of 97.6 percent.

Ahh, you're right!

Another source of loss could be the linear (viscous) drag term, which was set to zero in the simulation. I myself am not sure of the relative magnitudes of the linear and quadratic drag terms, if I can find my old notes on them I'll see about making a comparison.

Looking at the figure in Post #1, I wonder if the string could have been rubbing against the protractor?

 

[Naty1]

I didn't look at the math but was going to suggest the drag force is likely related to v²...but I see you have that...that's what the force of wind is on a boat...or other surfaces...(and helps determine appropriate anchor size)

I wonder if you changed the length of the pendulum (string) and compared actuals to your model if things would get closer or further apart...

 

[Redbelly98]

Update: I have run a simulation using Justanobody's parameters, and get the same 33% of initial energy after 300 seconds.

 

[Just a nobody]

Wow! Lots of replies. Thanks, everyone.

Hello just a nobody.The vibrating pendulum sets the support structure into vibration as well.Depending on how well this is constructed and set up it can be a major source of energy loss.

That's a neat idea. I'll try adding in a model for the support structure—do you think using the http://en.wikipedia.org/wiki/Shear_modulus" would make a decent model for this? In real life, I realize the rod in the support structure might actually bend (and not just shear), but that seems like it might be pretty difficult to model.

[...] In addition to what Dadface says, there might be drag on the string itself that you could add in, and there might be energy losses from flexing the string back and forth. You might try fastening it to a more rigid support to eliminate the losses Dadface refers to, and using some sort of bearing at the pivot point to minimize those losses. Another thing you could try is to use multiple bobs of different areas and plot the energy loss as a function of area, then extrapolate back to zero area. Then you could quantify how much of the energy loss comes from drag on the bob and how much comes from "other stuff". Then you could work to minimize the "other stuff"

I did try drag on the string (using the "wires and cables" coefficient from http://en.wikipedia.org/wiki/Drag_coefficient#General" and taking the thread to be 0.5 mm thick), but it was so insignificant that I took it out for the sake of simplicity.

The modifications to the experiment are all good ideas. Unfortunately, I no longer have access to the materials used for the experiment. My class is on Tuesday, so I'll check if the materials for last week's lab are still in the back of the room. If they are, I could probably get data for a couple other bobs at the end of the class period (assuming the TA doesn't mind).

Ahh, you're right!

Another source of loss could be the linear (viscous) drag term, which was set to zero in the simulation. I myself am not sure of the relative magnitudes of the linear and quadratic drag terms, if I can find my old notes on them I'll see about making a comparison.

Looking at the figure in Post #1, I wonder if the string could have been rubbing against the protractor?

I wasn't really sure what to use for the viscous drag term. Air isn't very viscous though, is it?

The string wasn't rubbing against the protractor. It just looks like that because of my bad drawing skills :)

I wonder if you changed the length of the pendulum (string) and compared actuals to your model if things would get closer or further apart...

According to my current model, a longer string should lead to more energy loss (a lower final energy/initial energy ratio). Unfortunately, I no longer have access to the materials for the experiment, so I can't really try that out in real life.

Update: I have run a simulation using Justanobody's parameters, and get the same 33% of initial energy after 300 seconds.

Thanks for checking it!

 

[Redbelly98]

I wasn't really sure what to use for the viscous drag term. Air isn't very viscous though, is it?

You're right. I've added viscous damping to the simulation, and it didn't change the results very much, dropping the final energy from 33% to 32%. I used a coefficient of 3e-6 kg/s, based on a presumed size 2 cm for the bob and the viscosity of air (1.8e-5 kg/m*s)

Thanks for checking it!

You're welcome. Thanks for writing understandable code!

 

[arildno]

Well, where is your friction at the hinge??

That is:
As the string (bound tightly around some axle?) will slide and shift, there will be quite a lot of energy loss there as well, not just energy loss due to air resistance.

 

[Jack21222]

Well, where is your friction at the hinge??

That is:
As the string (bound tightly around some axle?) will slide and shift, there will be quite a lot of energy loss there as well, not just energy loss due to air resistance.

This is the first thing that jumped out in my mind as well.

 

[arildno]

Now, a modelling suggestion for torque produced by friction at the axle:

I base my idea on the assumption that the string is "tightly" wound up around a stationary axis. It will still slide back and forth a bit.

Assume a Coulomb law, letting the string tension substitute for the normal force.
If "r" is the radius of the axle, you should have a damping torque as follows:
$$\tau=r\mu(mg\cos\theta+ml\dot{\theta}^{2})$$
where the frictional coefficient is between the material of the string and the metal of the axle.

 

[JolileChat]

Can you do the experiment again?
This time, try an initial condition half the original one.
Let us know the results.

 

[Just a nobody]

I'm going to return to this in a couple days; I'm a bit busy with homework right now. (So don't think I'm ignoring anyone's responses—I'm very grateful for all of your help!)

Well, where is your friction at the hinge??

That is:
As the string (bound tightly around some axle?) will slide and shift, there will be quite a lot of energy loss there as well, not just energy loss due to air resistance.

This is the first thing that jumped out in my mind as well.

The string isn't fastened like that. Sorry if my drawing was misleading. I attached a close up of where the string attaches to the support rod.

Can you do the experiment again?
This time, try an initial condition half the original one.
Let us know the results.

I'll find that out tomorrow. If the materials for last week's experiment are still in the room, I'll definitely try this out after class.