Dot Physics

Isn’t This Physics and Not Computer Science?


My new plan in introductory physics is to require students to demonstrate some ability to create a numerical calculation. Just to be clear, by “numerical calculation” – I mean to solve a problem by breaking that problem into many simpler problems. Typically, these problems are broken into small time steps and a computer is used to do all the boring calculations.

Previously, I have emphasized the idea of numerical calculations in class and even encouraged students to create their own. Starting now, that plan has changed. For me, it is no longer an optional topic. It is right in there along with the Momentum Principle and the Work-Energy Principle. Well, it isn’t really a principle, it’s more like a tool. I think that leaving numerical calculations out of an introductory physics course is like leaving calculus out of the same course. Sure you can do it, but it’s better if you include it.

In the past, there have been larger barriers for students to get into a numerical calculation such that many faculty just say “oh well”. This is no longer true (in my humble opinion). There are super simple tools like VPython and GlowScript (both excellent options). So there is no more excuse. And yes, I am talking to both students and faculty. Numerical calculations are just part of the way things are done. It would be a shame for students to never get a chance to practice this method.

So, how do I do this? Well, I have previously posted my notes on how I grade (Standards Based Grading mostly). The class mostly consists of short in-class assessments for which students can submit video screencast reassessments and this has been working very well. Now I just add a new assessment that is not connected to an in-class quiz and it is this:

  • Create a numerical calculation to solve some problem. You can use VPython, or GlowScript or really anything. I give several examples and a tutorial using VPython.
  • The student will get 3 out of 5 points if they create and show a program that models the motion of an object with a constant force (like a fan cart on a track or a projectile motion problem).
  • The student will get 4 points for a numerical calculation that involves a non-constant force (like the Earth orbiting the Sun).
  • Finally, the students get a maximum score of 5 out of 5 if their numerical calculation is for some type of problem that can not be easily solved analytically (like the 3 body problem or projectile motion with air resistance).
  • Once the students turn in the assessment by the due date, they can continue to improve their code over the semester to increase their score.

That’s it. Of course, there are still some problems. Some students wait until the last minute and then have some type of technical difficulty. Other students spend time to create a program, but it ends up just being some type of input-output calculator instead of a numerical calculation. But overall, it’s great fun. Students come in with interesting ideas and the discussions almost always revolve around physics ideas instead of coding and syntax errors.

But WAIT! I can’t do this because I’m not a computer scientist.

This is wrong. This numerical calculation is not computer science. When I think of computer science, I think of creating and exploring new algorithms in computer code. Creating a numerical calculation is just coding. Computer science is much more than just making code.

It’s also wrong another level. Student’s don’t have to be experts at computer coding to write a program. In fact, the great thing about humans is that they don’t JUST do ONE thing. Instead, they do all sorts of things. Here are some other comments about physics classes.

  • You don’t have to be an artist to draw a free body diagram.
  • You don’t have to be an author to write an essay that explains your physics problem.
  • You don’t have to be a mathematician to evaluate an integral of velocity to find the change in position.
  • You don’t have to be a public speaker to share your physics solution with the rest of the class.
  • You don’t have to be a physicist to like physics.

I can just imagine the first time a physics faculty told a class that they needed to draw a free body diagram of the forces on an object for the physics solutions. I wonder if a student complained that this was supposed to be a physics class and not an art class.

Nature of Science and Numerical Calculations

The last time I wrote about computational science I made the claim that numerical calculations are not any different than a theoretical calculation. Some people weren’t so happy with this idea.

Let me just add a couple of ideas about this. What is science all about? It’s all about building models. Well, it’s not only about building models. In order to call it “science” we have to see if that model agrees with real life. This checking for agreement is called an experiment.

I think most people will agree with my above definition of science. If so, you have to ask yourself “where do I put numerical calculations”? Are they experiments? Are they real life? Are they models? Well, they sure as heck aren’t real life (except for Flappy Birds – that’s real). If a numerical calculation doesn’t agree with real life, we like to call it something else. Suitable names would be either “art” or “computer game”. If the calculation does agree with real life, we call it a model.

In the end, students are mostly happy

Yes, many students had problems along the way to create their first numerical calculation. They started off confused and intimidated. But for those that pushed through these problems, they came out with their own creation. They had a program that ran and that they were proud of. It’s great fun.

Do Uniforms Make a Difference in Speed Skating?




Constant Power Humans




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Constant Drag Coefficient Humans





Angry Birds: How Do You Change Red to Pink?

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Images from Wikimedia Commons and Twitter

Surely, you know of Peter Vesterbacka. Right? Well, it seems like he always is wearing his signature Angry Bird red hoodie. Well, I was a bit surprised to see him with a Pink Stella hoodie on as he posted on twitter. This lead to the following reply from Peter:

CHALLENGE ACCEPTED.  So, what physics could I use to say that Peter’s hoodie changes from the color red to pink?

What is Color?

Maybe I should change that title to “how do we see color?” Every thing that we see is a result of two possible things. Objects can emit light (like a lightbulb) or objects can reflect light (like an apple). You are probably reading this blog post on a computer screen. In that case, there is most likely a light behind your display that emits light. That light enters your eye and your brain interprets the different wavelengths of light as color.

If you had this post printed out on actual paper, you would see it because light reflects off the paper and then enters your eye. For this second case, if you turn off all the surrounding lights there would be no light hitting the paper. This means that no light reflects off the paper and you couldn’t see it.

It doesn’t matter if an object emitted the light or the light reflected off the object, the color your brain interprets is based on the wavelength of the light. Of course, our human eyes can only see a small range of light that we call the visible spectrum. You can see this spectrum on your own by taking one of these inexpensive prism glasses. When light passes through them, different colors bend different amounts so that you can see all the colors.

Here is an image using my phone’s camera to look through one of these special glasses.

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Image: Rhett Allain

If you think you see pink in this spectrum, you could be wrong (but I think I see something pinkish in there). However, this image was created with a camera so that sometimes you don’t always get what is actually there (maybe). The point is that there a range of colors that correspond to different wavelengths and pink isn’t one of them.

How do we see the color pink? If you take all of these colors of light together, your brains says that it is the color white. If you just take the red and the blue, your brain says “PINK”. The same is true for other colors like “brown” and “olive”. They are all combinations of wavelengths of light.

What about the Doppler Effect?

This was my first idea of changing the color of a hoodie. It turns out that when an object moves towards or away from an observer at very high speeds, the apparent wavelength of light shifts. An object moving AWAY from an observer makes the wavelength of light longer than if it were a stationary source. Since red light has a longer wavelength than other colors, we call this effect a red-shift.

If an object was moving towards you at a very high speed, the opposite would happen. The wavelengths of light would be shorter and make it appear more blue. Here is my attempt at showing a green object that is both stationary and moving.

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So that is the Doppler effect with light. It’s a real thing and it is used in astronomy to measure the speed that stars are moving away from us. It can also be used to measure the wobbling motion of a star caused by a planet orbiting it.

At first glance it seems that the Doppler effect couldn’t be used to change Peter’s hoodie from red to pink. Sure, he could be moving away from us at really high speeds, but that would shift the color of his hoodie to longer than red wavelengths (infra red). If he was moving towards us, it could appear different colors, but it would have to be one of the colors of visible light. He could be moving very fast towards us such that the he appeared green or blue, but not pink since you can’t shift the red wavelengths to something that is a combination of wavelengths.

Really, the only way I can think to make the Doppler shift turn Peter’s hoodie to the color pink is if part of him is stationary and part of him is moving away from us really really fast. The part that moves away would be blue shifted all the way to a blue wavelength.  The rest of Peter would still have a stationary and red hoodie.  For the observer, there would be a combination of red and blue wavelengths making the hoodie appear pink.  However, there is one problem.  Breaking up Peter into parts is a bad idea.

If you would like another view of the color pink, check out this interesting video from Minute Physics.

But maybe Peter just took off his red hoodie and put on the pink one. I know that’s a boring explanation, but it’s one we should consider.

Just in case you want to read some more about light, here are some related links for you.

How Fast Can a Skater Turn in the Speed Skating Short Track?

In the Olympic event of Short Track Speed Skating, athletes race around a rather short track (thus the name of the event) that only has a circumference of 111 meters.

It always seems almost magical the way the skaters lean over so far while going around the turns. The faster the skater goes around the turn, the more he or she leans. Clearly, the skater can’t lean over 90°, right? So what is the maximum speed?

Here are some of the basic physics ideas:

  • Since the skater is turning in a circle, there must be a force pushing on the skater in the direction of the center of the circle. This force is the frictional force on the side of the skate.
  • If a skater didn’t lean at all, this frictional force at the feet would cause the skater to tip over.
  • By leaning, the skater can balance the torque from gravity with the frictional force. This seems like it would work, but I think you have to also consider the acceleration while turning.
  • I suspect that at greater lean angles, there is less frictional force on the skate.

There is one other big idea to use here. Fake forces. Yes, they can be dangerous and many introductory courses explicitly tell you they are bad – but they can be useful. What is a fake force? Well, the normal ideas of force and motion say that forces change the momentum of an object. But these rules only work if the reference frame is not accelerating (called an inertial reference frame). If you are moving along with a turning skater, you are in an accelerating reference frame. You can still make the force rules work, but you have to add a fake force.

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You are already familiar with fake forces. You use them all the time. When you are sitting in your car and push the gas pedal, the car accelerates forward. Since you are inside the car, you are in a non-inertial frame. So, what pushes you back into your seat as you accelerate? The answer: nothing. You can pretend like there is a force that is equal in magnitude to the force of the seat pushing you forward, but this is a fake force. In general, you can write this fake force as:

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Ok, back to the turning skater. Here is a force diagram including the fake force.

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So, just 4 forces. For an object in equilibrium (which would be true in our non-inertial reference frame), the following must be true:

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The net force in both the x- and y-directions must be zero as well as the torque (which we can treat as a scalar here) about some point. Here, I will use the scalar definition of torque about some point o:

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Where F is the applied force, r is the distance from the force to the point “o” and θ is the angle between F and r. If you want to know more about torque, this older post might be useful.

Oh, one more thing. What about the fake force? This depends on both the mass of the skater and the acceleration of the frame (which is also the skater). Since the skater is moving in a circular motion, the magnitude of this fake force would be:

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Here v is the magnitude of the skater’s velocity and R is the radius of the circle the skater is moving in. Now, there is a trick here. If the skater is leaning in towards the center of the circle, different part’s of the body will be at different distances from the center. If the radius is large enough, these distance differences won’t really matter. For the rest of the calculations, I will assume this fake force acts at the center of mass of the person.

Now I can start to put in some values. If we look at the total torque about the point where the skates touch the ice, I can ignore both the normal and frictional forces since they produce no torque. Oh, let me assume that the center of mass is in the middle of a skater with a height of h. This gives:

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This says a few things:

  • The faster you go, the more the skater leans (smaller angle).
  • Skater mass doesn’t matter.
  • The height of the skater doesn’t mater either.

How about a plot? We need a value for the radius of the circle. It seems like this is around 8 to 8.5 meters (depending on where the skater turns). I will go with a value of 8 meters. Here is the lean angle as a function of speed.

I suspect you really can’t get much lower than a 20° angle when leaning. That would put the maximum speed at about 14.7 m/s (or 32.9 mph). That’s probably faster than the skaters normally go – but let me just check. The world record for the 500 meter track is a time of 39.937 seconds. This would give an average speed (assuming the skater actually went 500 m – which probably isn’t true) of 12.5 m/s. Based on this value, I am going to put the best lean angle at 35° for a turning speed of about 10.6 m/s (and then the skaters could go faster on the straight away).

But what about friction? In this model, the friction force doesn’t change as the skater leans. If I assume the typical friction model that says the frictional force is proportional to the normal force, then we need to look at hte normal force. There are only two forces in the vertical direction: the normal force and the gravitational force. These must add up to zero and don’t depend on the lean angle. So based on this, the maximum frictional force is just some value. We can use this to find the coefficient of friction between the sideways pushing skate blade and the ice since the friction force must be the same magnitude as the fake force. Note that this model of friction probably doesn’t work for a blade cutting into ice.

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With a radius of 8.0 meters and a speed of 14.7 m/s, I get a coefficient of static friction with a value of 2.76. Typical values of the coefficient are usually between 0 and 1. So, this seems a little crazy. However, there is another way to get a value for the coefficient of friction between the skate blade an ice (for perpendicular motion). When the skaters start from rest, they use the side of the blade to push on the ice. This frictional force is what causes them to increase in speed. By measuring the acceleration, you could get another estimate for the coefficient of friction.

I will save that for either another post or you can do it for homework. Hint: use video analysis.

How Much Would a Real Olympic Gold Medal Cost?

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These obviously aren’t Olympic medals. Image: Rhett Allain

In some news item, it was pointed out that an Olympic gold medal is made with only 1% gold. Ok, that makes sense. A full 100% gold medal would probably be quite expensive. Also, gold is a rather soft metal so it might get bent or scratched too easily.

Let’s get an estimate of the cost of a fully gold Olympic gold medal. To start, we need some of the dimensions of a medal. Thankfully, Wikipedia has a nice page with both the size and mass of medals for different Olympics years (of course they do). From this, the 2014 Olympic Gold Medal has 1 100 mm diameter and a 10 mm thickness with a mass of 531 grams.

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I will assume that this is just a plain cylinder (which it probably isn’t since it probably has some raised features to it). The volume and density for this medal will then be:

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Using the listed dimensions, this gives a density of 6.76 grams/cm3. This is much lower than the density of gold with a value of 19.3 g/cm3.

If Not Gold, then What?

Now for the fun part. Let’s say that the medal is 1% gold (by mass). What is the density of the rest of the stuff? To start, I will write the total mass of the medal as:

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Hopefully it is clear that the m subscript represents the medal. I already know the mass of this other stuff in the medal, but I need its volume. If I subtract the volume of the gold from the total volume, that will just be the other stuff.

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Now I can solve for the density of the stuff.

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That’s it. It’s all finished except for the numbers. If I put the values from above into this expression, I get a stuff density of 6.717 g/cm3. Based on this list of metal densities, it seems like that stuff could be several things. It’s closest to the density of Cerium – but Wikipedia lists it as being both soft and it oxidizes in air. It could be cast iron or zinc. Modern pennies have a zinc interior, so maybe the medal is just like a huge penny.

Cost of a Gold Medal

The gold medal has a mass of 531 grams. If it was 100% gold, it would have a mass of 1.52 kilograms. Damn. That’s 3.35 pounds. You would surely feel that sucker around your neck. Wouldn’t you?

Ok, but what about the cost? In my estimate of the gold that Smaug has under the mountain, I used a gold price of $49.5 per gram of gold. This would put the medal at a price of (just for the gold) $75,992.5.

Next question: why don’t they make Olympic Gold Medals out of gold? I think the answer is obvious.

A Historical Look at Medals

Sometimes I can’t stop. Using the data from that Wikipedia page, I can plot the medal volume as a function of year (for both Winter and Summer Olympics).

Why did I add a linear fit? I guess just because I could. However, you can use it to answer the following questions:

  • If both the Winter and Summer Olympic medals continue this trend, how big will the medals be around the year 2050?
  • Could you get some type of exponential model to fit this data better than a linear fit?
  • Why do the medals seem to get bigger over time?  Is bigger better?
  • I wonder if there is any correlation between medal size and location of the Olympics.

Go ahead and play with it. Plotly lets you try your own fits and make this into your own graph. That’s what makes Plotly so awesome.  Oh, I left off a couple of medals that weren’t cylindrically shaped.

What about the medal density. There doesn’t seem to be a trend for the density so I will just plot it as a histogram.

Look at the medal for Summer Olympics in 1912 with a density of 18.26 g/cm3. That looks close to pure gold. Or maybe it’s mostly lead.

One more homework question.  How much would the 2014 medal be worth if it were made of Bitcoins?  Yes, I know that Bitcoins aren’t really real.  That just means that you have to be creative to come up with an answer.