Rhett Allain is an Associate Professor of Physics at Southeastern Louisiana University. He
enjoys teaching and talking about physics. Sometimes he takes things apart and can't
put them back together.
This is typically the first topic in the second semester of introductory physics - the interaction between objects with electric charge. There are 4 fundamental forces that physics typically looks at:
Gravity - an interaction between objects with mass - wow, I don't have a post on the universal law of gravity?
Electromagnetic - an interaction between objects with electric charge.
Weak Nuclear - an interaction between (let me just say for simplicity) leptons.
Strong Nuclear - an interaction between hadrons.
I know those last two are complicated - but I am not going to talk about the strong and weak forces.
Coulomb's Law
Coulomb's Law is a model for the forces between two charged particles. Here is the model.
Looks complicated, doesn't it? But that is the best way to represent that. Typically, books might just show the magnitude of the force, but I think you can handle the vector form. Let me draw a picture to go along with this.
Maybe you see the first problem - which force is this equation for? If the vector r is pointing from q1 to q2, then this gives the force that q1 exerts on charge q2. Note that it is important to include the unit vector r-hat in the equation or you would not have both sides of the equation as vectors. Also, the denominator is the magnitude of this vector squared. If you did not know the vector r, but instead had the location of the two charges (this happens a lot), then:
Also, in that equation, the 1/4pi-epsilon term is a constant. The charges should be in units of Coulombs, and the distances in meters - this will give a force in Newtons. Here is the value for that constant: (some texts just call this k or something)
One final note - in this form, the sign of the charge DOES matter, and this a good thing. If the two charges have opposite signs, the direction of the force will be in the opposite direction as the r-hat vector. If both charges have the same sign, then this force will be pushing the two charges away from each other.
Maybe this seems like an unnecessarily complicated form of Coulomb's Law, but I think it is the most useful in the long run.
Comparison with gravity
It is useful to make a comparison between the universal law of gravity and the Coulomb force. Here is the gravitational force:
This looks similar to the Coulomb equation. The big difference is the negative sign. This is there because gravity is always an attractive force but the mass is always positive.
Where does this come from?
This was a great question I had introductory physics one time. Actually, I think it was something like "how do you derive Coulomb's Law?" The answer: you don't. Coulomb's law is a mathematical model based on experimental data. The basic idea is to get two spheres and put some electric charge on them. Find some way to measure the force between them and see how that compares to the distance. There is a trick here - a uniformly charged sphere looks just like a point charge. Coulomb's law deals with point charges.
An Example
This is what my students would want - a worked out example. Here is my completely made up example. Suppose I have a 3 nanoCoulomb (nC) charged ball at the origin and a -7 nC charged ball at the location < 0.3, -0.4, 0 > m. What is the force on the negatively charged ball? Here is sketch:
The first thing I need is the vector r. This is pretty easy since one of the charges is at the origin. I also need the magnitude of the r vector and the unit vector r-hat.
Now, I just have to put stuff in. Here is what I get:
Not too bad, is it?
Superposition
One last very important thing. What if you have more than two charges? The superposition principle applies to the Coulomb force. Basically, this means that if you have three charges (called A, B, and C), then the force on C will be the vector sum of the force on C due to A and the force on C due to B. Here is a picture.
I will do an example with more than two charges in a later post.
The other day I was having a discussion with my brother about software and free software. Somehow I said I like it when software is free. Since he is a programmer and likes to get paid to program, he said "education should be free." I think I might agree with him.
I previously talked about higher education, specifically how it was more about being human than about job training. With that in mind, shouldn't it be free? If everyone should get healthcare, I think everyone should get an education.
But wait! How would I get paid? I haven't really thought this through the whole way. But there are some possibilities. Taxes? Does that count? Ok, how about patronage? That would be the best option. Another option would be sponsors. Faculty could pick up sponsors and wear the logos on shirts and stuff. If this happens, I would like Red Bull to be my first sponsor. Or maybe some coffee shop.
Oh, I know you missed it. Really, it wasn't your fault. Pi day fell on a Sunday, so how are you supposed to have pi-day activities in class? Don't let it stop you. You are better than that. Do the activity anyway. What to do? Here are some suggestions. (Suggestions aimed mostly at the high school level)
Plot Diameter vs. Circumference
This is a great one. Let your students find as many round things as they can (cylinders work the best - or flat stuff). Measure the circumference (you can use a string or a tape measure) and the diameter. Since the relationship between these two is:
A plot with circumference on the vertical axis and diameter on the horizontal axis should be a straight line with a slope of Pi. The great thing about this activity is that students can get a feel for where pi comes from. There are two many students that just think it is a number invented by mathematicians to make things more complicated and cool sounding.
Oh - as a bonus, students get to practice making graphs and finding the slope. I would recommend doing this one on real graph paper (and not in a spreadsheet).
Rolling vs. Distance
Really, this is the same thing as the activity above - but it looks different. Take a cylinder and roll it. Count the number of revolutions and measure the distance it rolled. Plot distance on the vertical axis and the number of revolutions on the horizontal axis. Here is the relationship between the two.
LEGO Estimation of Pi
I just thought of this one and have not actually tried it. Archimedes estimated Pi by drawing two 96-sided polygons inside and outside of a circle. He could then determine an upper and lower bounds for the value of Pi. You could try to reproduce this Lego pieces. Make an n-sided Lego polygon both inside and outside of a circle. Compare the perimeter of the polygons to the radius of the circle.
I might make this a future post, but if you try it out, let me know how it works.
Monte Carlo Estimation of Pi
I had a more detailed post about this method for estimating Pi. But maybe you don't want to look back - so here is the short version. If you randomly put points in a 1 x 1 square, some will be more than 1 units from the corner and some will be less. Here are some random dots.
Since the dots that are less than 1 from the lower left corner make up 1/4th of a circle, the ratio of red dots to total dots should be:
So, this is pretty straight forward. But how could you do this? I made a program in Scratch as well as python. You could use anything that has a random number generator. Here is a version in google docs:
I didn't finish it, well mostly I did - but you would need to do some more work on it to finish. If you wanted, you could have groups in the class calculate the average for 100 points and then take the average average for all the groups.
Non-Computer Monte Carlo
Maybe you think computers will one day rule the world and you would rather not use them to calculate Pi. I can understand that. You could drop something so that it has a random distribution on a 1 x 1 square and then count the number in and outside of a circle. Maybe find some way to drop sand on a square paper? Make sure that sand is falling outside of the paper also, or it likely will not be randomly distributed.
How Accurate can you get Pi by Measuring?
What if you used the plot of circumference vs. diameter from above? How accurate of a value of Pi could you get?
Other resources
There are tons of great Pi sites out there. Here are just a few:
Do you have an iPhone? I have an iPod Touch. Do you use vectors? Do you think RPN calculators are the bomb? Here is a RPN calculator for the iPhone that supports vector notation. VCalc by Silicon Prairie Ventures Inc. Oh - it is free. Here is a screen shot.
I played with it some, the only thing that would be cool would be a landscape mode.
The most basic explanation of Pi is that it is the ratio of the circumference to the diameter for a circle. That seems simple enough, but it turns out that Pi is an irrational number - so you can't just write it down. Oh, I know that you are an uber-geek and you could recite the first 80 digits of Pi. But the question is - how many digits are enough?
In this post, I am going to assume that we don't know the true value of Pi (which is essentially true). I can then use propagation of error techniques to see how dependent different calculations are on the value of Pi.
Super Brief Intro to Uncertainty
I still can't believe I haven't put a post together on the basics of measurement and uncertainty. Add that to the todo list. The most important idea in measurements is that they are not exact values. Let me start with my favorite example. Suppose I have a table that I want to know the area of. To do this, I measure the length and the width. The value I come up with for the length is 133.2 cm. But what does this mean? Is this the exact length of the table? No. Two problems.
The table doesn't have an exact length. What does the length mean for a table? Is it a perfect rectangle? No. Is it even straight on the edges - probably not.
Even if it were a perfect table, would my measurement be perfect? No.
Maybe I measured this length a whole bunch of times and at different locations. This would give me an estimate of how the measurements are spread out. If I do the same for the width, I might get something like:
This means that the length of the table is almost certainly between 133.0 cm and 133.4 cm. If a similar thing can be said about the width, then this diagram could represent the area.
The point I would like to make - since the width and the length have uncertainty, the calculated area would have uncertainty. How do you determine this calculated uncertainty? I have three ways:
Use the extreme values of length and width to calculate the extreme values of the area (in this case the smallest area uses the smallest length and width). This is the method I use for my algebra-based physics labs.
Assume the error is small, linear, and normally distributed. In this case, you can use the partial derivatives of the functions to determine the relationship of the uncertainty for the measured stuff on the calculated stuff. Here is wikipedia's page on this, but I am not really going to go into the details.
Assume that if you measure the stuff a whole bunch of times, the data would be normally distributed. Write a program that generates normal data and use that to calculate tons of times the calculated value. Look at the spread of all these calculations to determine the uncertainty. I am not going to do this right now.
Back to Pi
Archimedes used 96 sided polygons to estimate the value of Pi. He showed that Pi was greater than 3 and 10/71 and less than 3 and 1/7th. This gives a decimal value from 3.14084507 to 3.142857143 (with no rounding). I could write this as an average and an uncertainty of about:
That is not too bad of a value. But what about pi = 3? Is that bad? First - according to Snopes, no state has ever proposed a law that would officially change Pi to 3. It is still a fun story. Anyway, in this case I could perhaps say:
I chose the uncertainty in this fictional Pi to be +/- 0.2 so that the range would cover the true value of Pi. Really, though you could in general write Pi as:
Where Delta pi is the uncertainty in pi.
Some uses of Pi
So what effect does the uncertainty in Pi have on different uses of Pi? Let me start with something practical - the speedometer in your car. Basically, your speedometer needs Pi to make the conversion between angular velocity and linear velocity using:
I know, there is no pi in that equation. But, how do you know the angular velocity (omega)? If this is measured in revolutions per second (or minute) then you have to convert units. Let me write this as:
Now, I will assume that omega, r, and pi all have uncertainty. Then the uncertainty in the velocity would be (using the max-min method from above for simplicity):
And I would do a similar thing for the minimum value. I could average the difference between average and the max and the average and the min. (I will put these calculations in a spreadsheet for you).
What about the volume of a sphere? This same thing is used for calculating things such as - the volume of the sun or the volume of a spherical cow. Here is the volume of a sphere:
These two uses of Pi seem boring - but really this is the basis for many applications of pi. There are tons of others, but they are maybe more abstract (but just as important). Now, on the to the spreadsheet. I will put in some values for the stuff, but you can change them if you like.
Note - I don't know how to change the number of digits presented in google docs. Also, I seem to have hit a creative wall with uses of pi. How about you list your favorite use of Pi in the comments?
Click the image to run the scratch app. It is kind of fun to watch.
About Scratch
If you are not familiar with Scratch, basically it is a graphical programming language a lot like the stuff for the Lego robotics. It is free, runs on Windows and Mac OS X, and can be embedded in a webpage as a java applet. Sure, Scratch has some limitations, but it is great for kids. Here is what the code for the pi-estimation program looks like:
If you create an account on Scratch, you can download the code of any project. I like Scratch.
Pi day is March 14th - get it? (3.14) I am a big fan of Pi. Here is my first post to celebrate the awesomeness of Pi (I know this is early, but I was too excited to wait).
How can you determine Pi?
Oh sure, tons of high schools do the classic experiment. Measure the circumference and diameter of as many round things as possible. Plot diameter vs. circumference. The slope will be Pi. Really, this is a great lab to do for all sorts of ages. The key thing is that students can see what Pi really means. I am not going to talk about this lab, I am going to do some thing cooler.
What if I had a 1 meter by 1 meter square taped out in the grass near the physics building and shot ping pong balls off the roof at this square? Ping pong balls are very difficult to aim, so you could easily imagine that they would fall in a random pattern in the square. I don't really want to set this up, so I am going to do it with a program instead. Here is a program that will generate random points in a 1 x 1 square.
This is what the output would look like:
Now, here is the key. What if I calculate the distance of each of these point from the origin? This would simply be:
Doing this, I can separate all the points into those that are more than 1 unit away from the origin and those that are less than 1 unit away. In this plot, (of 1000 points) the red dots are more than one unit away and the blue are 1 unit or less.
Note that this plot does not have exactly the same horizontal and vertical scale - no idea why it came out like that. However, this is enough dots that maybe you can see a pattern. The blue dots are filling up 1/4th of a circle. The full area of this circle would be:
Since I am a physicist, I have trouble leaving the area as unitless. What if I look at the ratio of blue dots to total dots? This should be the same as the ratio of the area of the quarter circle to the whole square, or:
Doing this for my last random number run, I get pi = 3.04. What happens as I increase the number of random dots? This a plot of the estimation of Pi for numbers of dots starting at 1000 up to 100,000 dots. I did this 5 times because each time is different.
So, you can see that as the number of random points increases, the estimated value of Pi is a little less spread out and closer to 3.14-something.
P.S. Thanks to Dave for this idea. You know who you are.
Mario has appeared in over 200 video games as Nintendo's main mascot. He has starred in TV shows, comic books, and feature films, but anyone who's played one of his games knows it's not all fun and games for the lovable plumber. He is constantly falling down bottomless pits, getting eaten by giant mutant mushrooms, and otherwise killing himself all while trying to save a princess who is apparently missing the part of the brain that allows one to avoid getting kidnapped. With that in mind, your estimation question is this: "How many times has Mario died in all of the videogames ever played?"
If you are interested, head on over to Diary of Numbers and submit your answer.
So, the question is: is the motorcycle thing real or fake?
First, let me talk about the key aspect of this demo. Why don't the glasses move? Well, they move - but just not very far. The demo is supposed to be an example of Newton's Second law, or you could say it is the momentum principle (which is what I will use). If a force is applied for a short time interval, the momentum will not change too much. Here is the momentum principle:
And here is a diagram of a glass on a table with the cloth being pulled out.
Ok, so I know the demo works - I have done it. However, let me look at some of the parameters involved.
Static vs. kinetic friction
If the glass is moving with the same speed as the tablecloth, then there would be static friction between the two objects. This is bad because 1) the glass would be moving and 2) the static friction can have a greater value than the kinetic friction. The key is to make the table cloth have a high enough acceleration so that the static friction force is not enough to keep the glass at the same speed as the cloth. So, the magnitude of the max static friction force is:
Where N is the force the tablecloth pushes up on the glass (the normal force). Since the glass is not accelerating in the vertical direction:
Where m is the mass of the glass. So, the maximum horizontal acceleration for static friction would be:
And, this is why I put some slack in the cloth before pulling it out. I can get a huge acceleration that way.
Time interval
So, suppose it is sliding. No need to worry about the static friction force. The force diagram would essentially look the same except that the friction force would be a bit smaller. The next key thing is the time interval. You want this friction force to be on the glass for as short of a time as possible. The momentum principle says:
I guess I could put in the model for friction. In this case, I could get the scalar equation (for the horizontal direction)
Now for some measurements
This video has something very useful. First, the demo is done as a normal demo. If I assume this first part is real (and why wouldn't it be - since I can do this myself?), then I can assume that the parameters are the same for both cases. Using my uber-tracker video skills (Tracker Video), I can get the time for the cloth to leave and maybe an estimate for the motion of the glass.
I know this is really a rough measurement, but from that video - the cloth is moving under the glass for about 0.12 seconds. If I had a good measurement of the final velocity of the glass, I could get an estimate for the coefficient of friction. This is my best shot:
So, if I assume a change in velocity 0.19 m/s and a delta t of 0.12 seconds, then:
Measuring the BMW version
Whoever made this video hasn't paid attention to what I have said about video analysis. The camera is not on a tripod AND they zoom out. Well, let me first get the time. Going through frame by frame, it looks like it took about 0.6 seconds. I will give it a value of 0.5 seconds just to be safe.
So IF I assume that the coefficient of friction is the same, then how fast would the glasses be moving after the tablecloth is pulled out?
That gives a speed that is about 4 times faster than the other dishes. Those dishes do move, but it seems like they should move more than that.
Is it fake
So, here are the options:
It is fake. Don't know how they did it, but they did something. Whatever they did, it looks convincing enough. The dishes do move some.
It is not fake and the model I was using for friction does not apply in this case because the tablecloth is moving so fast (remember, friction is actually pretty complicated).
It is not fake and I just made an error somewhere - perhaps with my measurements.
Update
It seems like The Blog of Phyz posted this before I finished my analysis.
Rhett Allain is an Associate Professor of Physics at Southeastern Louisiana University. He
enjoys teaching and talking about physics. Sometimes he takes things apart and can't
put them back together.
