Experimental Derivations of Pi in Physics

[JacobPhys]

Greetings

I'm currently designing a lab-script intended for comprehension and use at an undergraduate level. I was extremely frustrated during my undergrad to be dealing with a plethora of uninspired and dull experiments so I decided to take a slightly unconventional (or extremely conventional depending on how you view it) approach to the task.

I've broken the script into 3 mini-sections, each hopefully touching base with a different area of Physics/Mathematics with a progressively detailed/complex experiment. The first section is using Buffon's needles (geometric probability) to determine pi and the second uses the age old SMH simple pendulum experiment that some of you may remember from first year/A-level (calculating pi instead of g).

For the third section, I was hoping to design/use an experiment that required a deeper understanding of the Physics taught in most undergraduate curricula but I'm somewhat at a loss. I've contemplated using Stoke's Law for viscosity but I wanted to be a little more creative. Something relating to the Uncertainty Principle has also come to mind but I'm at a complete loss as to how to quantify pi using said principle.

I was hoping that perhaps someone could offer some insight/ideas for a potential third part to this lab? I would go as far as to include simulations that employ programming (specifically in python or fortran95) if need be.

Thanks in advance

 

[Dale]

$\pi$ is a mathematical constant, not a physical constant. It can be calculated to any desired precision without requiring any experiment. If you want to do experiments about physical constants you should look into the fine structure constant as the quintessential example.

 

[JacobPhys]

$\pi$ is a mathematical constant, not a physical constant. It can be calculated to any desired precision without requiring any experiment. If you want to do experiments about physical constants you should look into the fine structure constant as the quintessential example.

That is unfortunate considering I've already submitted the idea and received approval from my superiors (they didn't think it was a wholly terrible idea). My hope was to provide pre-established constants that one would normally derive in these types of experiments and work from there. This is an undergraduate script so it's not so much about the results and data (though commentary on these is a necessary in the context of the experiment) but rather about the experimental practise and ability to analyse the experiments themselves (what physical factors in the experiment may lead to a lack of precision? etc)

 

[Dale]

So you basically just need some relationship between two or more physical quantities which contains $\pi$ where you can experimentally set all of the relevant parameters, measure the experimental numbers, and get $\pi$ out. And now specifically you want to use physical quantities related to the uncertainty principle.

 

[JacobPhys]

So you basically just need some relationship between two or more physical quantities which contains $\pi$ where you can experimentally set all of the relevant parameters, measure the experimental numbers, and get $\pi$ out. And now specifically you want to use physical quantities related to the uncertainty principle.

Well I thought it would be nice to use a slightly more complicated piece of Physics in the final part. Nothing beyond the scope of the regular BSc content, so something that employs the uncertainty principle or an idea that requires a similar level of comprehension would be ideal (not to say I'm limiting this part to tests of/in QM). I was just hoping for some potential ideas that could easily be conducted in a lab.

 

[fresh_42]

I thought that this could be a valid approach:
https://aip.scitation.org/doi/full/10.1063/1.4930800
Unfortunately it's behind a paywall.

 

[Dale]

Well, if you can do photon counting experiments in a laser with maximal coherence then there is an uncertainty relationship between photon number and phase. Those are usually expressed in terms of $\hbar = h/2\pi$, so you can solve for $\pi$

 

[JacobPhys]

I thought that this could be a valid approach:
https://aip.scitation.org/doi/full/10.1063/1.4930800
Unfortunately it's behind a paywall.

This is actually a fascinating little read, neat proof but I am slight hard-done-by as to how you would be able to bring this to a lab?

Well, if you can do photon counting experiments in a laser with maximal coherence then there is an uncertainty relationship between photon number and phase. Those are usually expressed in terms of $\hbar = h/2\pi$, so you can solve for $\pi$

I'll have a look to see if I can put together anything using this, unfortunately H&S are extremely reluctant to let undergraduates use lasers. I'll have to see if I can twist their arm a bit to do this.

 

[Dale]

unfortunately H&S are extremely reluctant to let undergraduates use lasers. I'll have to see if I can twist their arm a bit to do this

If you are not stuck on the uncertainty principle, then basically any quantum mechanical relationship expressed in terms of $h$ or $\hbar$ will work. You could get some blackbody spectra of known temperatures and use those to calculate it, or you could use the Josephson effect or the quantum Hall effect.

 

[PeroK]

That is unfortunate considering I've already submitted the idea and received approval from my superiors (they didn't think it was a wholly terrible idea). My hope was to provide pre-established constants that one would normally derive in these types of experiments and work from there. This is an undergraduate script so it's not so much about the results and data (though commentary on these is a necessary in the context of the experiment) but rather about the experimental practise and ability to analyse the experiments themselves (what physical factors in the experiment may lead to a lack of precision? etc)

 

[Dr. Courtney]

The period of a simple harmonic oscillator is given by T = 2 pi sqrt(m/k) where m is the mass on the spring, and k is the spring constant.

T, m, and k can all be measured fairly easily, so pi can be computed and considered an experimental result. Accuracy between 0.1% and 1% should not be too hard. Accuracy to better than 1 part in 1000 tends to be more challenging, as you may face issues with the non-zero mass of the spring and your spring constant changing with temperature. The mass on the spring will tend to do better than the simple pendulum, because of challenges independently measuring g and the pendulum length to 1 part in 1000.

Thinking of simple physics experiments to measure pi is not too hard. The more challenging idea is adding accurate digits to pi after the first 3-4.

 

[Andy Resnick]

Greetings

I'm currently designing a lab-script intended for comprehension and use at an undergraduate level. I was extremely frustrated during my undergrad to be dealing with a plethora of uninspired and dull experiments so I decided to take a slightly unconventional (or extremely conventional depending on how you view it) approach to the task.

<snip>

I was hoping that perhaps someone could offer some insight/ideas for a potential third part to this lab? I would go as far as to include simulations that employ programming (specifically in python or fortran95) if need be.

Thanks in advance

I applaud the intent, but share Dale's critique of your approach. In any case, since π is defined as the ratio of a circle's circumference and diameter, why not have the students 'measure' π for non-flat coordinate systems? (for example, determine the ratio of circumference and diameter for a circle drawn on a ball or on a Pringle's potato chip).

 

[haushofer]

I really liked the new 3blue1brown video, about how you can determine pi with 2 colliding blocks. I'm not sure how practical it is, but it is a beautiful constructing in dynamical systems.

 

[Dr. Courtney]

I really liked the new 3blue1brown video, about how you can determine pi with 2 colliding blocks. I'm not sure how practical it is, but it is a beautiful constructing in dynamical systems.

Great video, but I regard it more as an exercise in theoretical elegance than experimental practicality. No way to get collisions sufficiently elastic or surfaces sufficiently frictionless to get the experiment to work. A quantum or electrical analog might be more practical.

 

[haushofer]

Great video, but I regard it more as an exercise in theoretical elegance than experimental practicality. No way to get collisions sufficiently elastic or surfaces sufficiently frictionless to get the experiment to work. A quantum or electrical analog might be more practical.

Yeah, a black hole threatens to form beyond a certain accuracy, let alone the question of friction. But it would be fun to see how accurate it can get :P

 

[fresh_42]

Yeah, a black hole threatens to form beyond a certain accuracy, let alone the question of friction. But it would be fun to see how accurate it can get :P

An air hockey table could be helpful ...

 

[Dr. Courtney]

An air hockey table could be helpful ...

Several challenges there.

1. Even with a coefficient of drag of 0.01, the circle that represents the phase diagram in the video is going to be shrinking significantly.
2. The experiment requires confining the motion to 1 dimension. Means of doing this will invariably increase the friction.
3. True elastic collisions are rare with things like real balls and hockey pucks. I can't think of any real mechanical collisions that retain more than 99% of their initial energy.

So, taking a 99% energy retention as likely the best available, the circle in phase space is going to be a lot smaller by the time you have the 31 collisions required for 2 digits of pi and vanishing by the time you have the 314 collisions required for 3 digits (and this is ignoring friction).

I think you'll need a much more clever design than an air hockey table to even get the 2nd digit.

 

[fresh_42]

I think you'll need a much more clever design than an air hockey table to even get the 2nd digit.

Probably true. I just thought it would be fun. However, the classical experiment: "throw a match on a lined paper" is probably not much better and a lot more boring. On the air hockey table one could at least count the pings with a micro, whereas an automatic counting on the paper takes more sophisticated hard- and software.

 

[Dr. Courtney]

Probably true. I just thought it would be fun. However, the classical experiment: "throw a match on a lined paper" is probably not much better and a lot more boring. On the air hockey table one could at least count the pings with a micro, whereas an automatic counting on the paper takes more sophisticated hard- and software.

A quick look at the Newton's cradle literature suggests a modified Newton's cradle design has a better chance of getting the second digit of pi (31 collisions). A Newton's cradle design has several advantages:
1. Motion is confined to 1-D without introducing large losses with additional friction.
2. Collisions can be nearly elastic.
3. No sliding friction, just air resistance.