Window's Vista

This week, Futility Closet had a post about the Vista Paradox. The paradox is connected to an effect I discussed before, where the angular nature of our vision causes objects to move in unexpected ways. In this case, we're viewing a distant tower through a window. As we approach the window, the window takes up proportionately more of our vision that the tower, causing it to appear to shrink. I tried making an animation of this to test my understanding:

As expected, the tower (red) changes very little, while the closer window (blue) expands rapidly. Initially, I was concerned with how the window's proportions appear to change, but I checked with my mother Sally, and she gets the same expression I did. There still may be an error in my implementation though.

After reading the Futility Closet post, I was reminded of an effect often used in filmmaking called a Dolly Zoom: The camera moves forward (backward) while zooming out (in), causing the edges of the frame to stay where they are, but the center to contract (expand). The scene that always comes to my mind for this is one from Fellowship of the Ring:

makeagif

Using the same framework as above, I plotted a series of rings at different distances, but the same size. Then I moved the camera toward them, while fixing the edges of the plot to the nearest ring:

This effect makes the center appear to shrink, the opposite of the motion in the film, but exactly the same technique. It's neat to see this artifact of human perception appear in both architecture and filmmaking!

Gone Fission

This week, Marika and I watched HBO's Chernobyl series. In spite of the depressing topic, and the at times gruesome imagery, we found it extremely interesting. I was (of course) most drawn to the explanation of how the reactor operated, and then failed. I've never studied nuclear physics in detail, but I had a rudimentary understanding of how reactors work:

• Uranium atoms radiate particles which can hit other uranium and cause them to split
• When uranium splits, it releases energy, which heats the fuel
• The fuel is cooled by water, which drives a turbine to generate power
• Control rods block the radiated particles, slowing down the reaction

The final episode of the series shows the trial of the technicians in charge of the plant, and a scientist gives an excellent explanation of how their decisions and flaws in the design caused the disaster. There are a number of interconnected systems that control the behavior of the reactor, including the control rods and the cooling water, but also buildup of gasses and the power output of the reactor when any of these properties change.

I was curious if I could create a rudimentary simulation of these systems to see if I could cut it as a Soviet reactor tech. I decided to base my model on a boiling water reactor, which seemed to be nearest to my previous understanding of how these plants work. A bit of searching turned up this document describing exactly the type of setup I was imagining, intended for training power station operators!

I wasn't interested in reimplementing a product developed by the International Atomic Energy Agency, but their description was a good starting point to identify the variables, and pare down the complexity. The elements I chose were: Reactor power output P, which increases or decreases as the control rods are removed or inserted; the heat transferred to the water ΔQ, which corresponds to the electric power produced by the plant; the volume of water surrounding the fuel V; the rate water is pumped into the reactor F; and the temperature of the fuel T. These are related by some differential equations. The heat transferred from the fuel to the water is proportional to the temperature of the fuel, and how much water is in contact with it:

That heat is taken out of the fuel, which cools it, while the nuclear reactions I described above are heating it:

The heat is also causing water to evaporate, but we're pumping water in to compensate:

I've played a bit loose with the units here – There should be conversion factors to go from energy to temperature, and energy to volume. I just wanted to get a feel for how these parameters interact, and I did! Below, you'll find another HTML5 simulation, where you can control how far the control rods are inserted, and how much feed water is entering the reactor. If the reactor gets too hot, it will meltdown, and you'll need to reset. I don't think it's very realistic, so aspiring nuclear technicians should not include this on their resume, but you can see some interesting behavior. Have fun!

Ezra & Phineas & Phineas & Ferb

This week, my nephew Ezra had a question about a movie he watched with his brother Phineas, Phineas and Ferb the Movie: Candace Against the Universe. In the movie, the main characters' sister is taken to an alien planet, which is surrounded by an "ion barrier".

There are two scenes that involve the ion barrier, and Ezra wanted to know whether either made sense, physically speaking. In the first scene, two groups of characters try to build portals to the other planet, but find,

There is an ion barrier around the planet. Our transporters were both deflected, which made them connect to each other.

Before we get into whether that makes sense, let's talk about what the word "ion" means. Atoms are made up of a nucleus of positively charged protons and neutrally charged neutrons, surrounded by a cloud of negatively charged electrons. Usually, there are an equal number of protons and electrons, which makes the atom neutral (no charge). However, sometimes it can gain or lose one or more elections, "ionizing" it. My guess is that the ion barrier is a layer of charged atoms surrounding the planet.

So, how could such a setup deflect a transporter? Well, assuming the portals used electromagnetic waves to transmit people, this might actually make sense. Electromagnetic waves include oscillating (moving back-and-forth) electric fields, which cause charged particles to oscillate as well. Those oscillating charges then create their own electric field, which can generate new electromagnetic waves. This is actually how mirrors work: The metal surface of the mirror is full of electrons that can move around freely – That's what it means to be a metal. When light hits those electrons, they reemit that light in the opposite direction, reflecting the image.

Supposing one transporter sent an electromagnetic wave, it could be reflected by this ion barrier, and be received by the other one. Unfortunately, the universe is a big place, and even going at light speed it would take 8 years to make this round-trip to the nearest star. They don't say anywhere in the movie how far away the planet is, but in any case their sister would be waiting an awfully long time!

In place of the transporter, the characters use a rocket ship to get to the planet, but once again encounter problems with the ion barrier:

So, if we go through the ion barrier without a shield, it could fry all of the electronics on the ship, rendering our navigation useless and stranding us in space! [...] In episode 206B of Space Adventure [the show's Star Trek stand-in], they were able to go through an ion barrier without a shield by spinning the USS Minotaur and scattering the ions as they went.

The first part of this makes sense: Charged particles, like the electrons in wires and the ions in the barrier, attract and repel each other. That means that while moving into the cloud of ions, the electronics in the spaceship could experience more current than they were designed for and fail (though likely not as spectacularly as the exploding control panels of Star Trek, and presumably Space Adventure). However, as far as I can imagine, spinning would make no difference, nor would "scattering" the ions, since the electric field they produce would still be present to disrupt the electronics.

As is often the case, storytelling is favored over scientific accuracy. While I nitpick the physics that appears in pop culture, I'm generally happy to suspend disbelief for the sake of a good story, and I'd say this movie certainly qualifies. Thanks for introducing me to it, Ezra!

Gold Standard

My brother-in-law Alex is a fan of anime, and recently I saw him watching One Piece, a show I wrote about long ago. Since making that post, I learned about a better model for the situation, and I thought I'd revisit it, and see whether the writers know their statistical mechanics!

As a reminder, the main character in the show, Monkey D. Luffy, is described as a Rubberman. Rubber is a type of polymer, which means it consists of a chain of repeating units, called monomers. As we physicists often do, we can take the absolute simplest form of this: Each link of length a either goes up or down, with equal probability.

We can find the length of the chain and the total number of segments in terms of the number that go up, and the number that go down:

What's interesting about this model is that these equations do not specify specific links in the chain, only the total number that go up or down. That means we have a system with indistinct microstates, and entropy becomes relevant. For a system like this, the entropy is given by

What does all of this have to do with stretching though? For that, we turn to the first law of thermodynamics:

where U is the internal energy of the system, dQ is the heat added to the system, and dW is the work done on the system. For our system, we want to keep the energy constant, so we can set dU = 0, and the second law of thermodynamics gives dQ = T dS. The work done on the system is a force applied by the chain multiplied by a displacement, or dW = -f dL. The work is negative because the chain pulls in the opposite direction it's stretched. Putting all this together, we get

To find this derivative, we can solve the first 3 equations together to get S in terms of N and L. Skipping all the algebra involved, we end up with

where L0 is the length with no force applied. To find a and N, we can look back at the diagram from last time:

The sphere's mass is 29 million kilograms, and we can multiply by g to get the force. The temperature is around 300 K. We can get L from the diagram, and according to DaVinci, Luffy's unburdened arm span L0 should be the same as his height.

Wikipedia

Rearranging the equation above,

Plugging in the values, B = 26.8 m, but A = exp(1.3729469e+29 m^-1), which is an absolutely enormous number. That suggests that for a and N to be close to the same order of magnitude, we would need around a quadrillion links, each on the order of femtometers. I thought maybe a more detailed analysis would make this situation a little more understandable, but that ball is just too damn heavy!

Alright Guv'nor!

The other night, we watched the movie Mortal Engines, about a post-apocalyptic society that lives in steampunk-style mobile cities:

YouTube

All the retro-futuristic equipment got me thinking about one of the more popular devices to show, centrifugal governors.

Wikipedia

Some of the steam driving an engine is diverted into this device to make it rotate. The faster it goes, the more the arms swing outward. Depending on the angle, more or less steam is fed into the engine. I was curious to look at some of the physics behind these tools.

We can start by looking at just one of the arms:

The rotational inertia for this design is

which gives the rotational kinetic energy

Meanwhile, the height of the weight is

which gives the gravitational potential energy

We can combine those equations to relate ω and θ, and then use it in the torque equation:

In the case of a steam engine, we want to release pressure if the engine is going too fast, so we could imagine making the torque proportional to the height of the weights by connecting a valve to the arms. Then we get a differential equation for the rotation speed:

where A is the constant of proportionality. If A is negative, then high speeds get slower. As is, this equation would drive toward zero, but it could be engineered to have some minimum speed. I recognize it can get a bit silly at times, with useless gears and cumbersome designs, but I still have a soft spot for steampunk aesthetics, so I enjoyed the film.

I Have a Bad Feeling About This

I often have ideas for this blog that aren't quite developed enough to make a post, so I put them on a list to come back to later. This one is far from the oldest, but it is from 2015: Starkiller Base from Star Wars VII: The Force Awakens. What struck me was the details they showed:

The weapon is powered by siphoning off energy from a star, and then blasting it at a distant target. It occurred to me this week that it's actually pretty easy to figure out the energy needed to blow up a planet, and so I could assess the feasibility of doing it with power from a star. (Marika is sure I'll be put on government watch lists for talking about this.)

Planets (and stars) are held together with gravitational potential energy, which depends on the mass and the density. You can calculate it for a uniform sphere by adding up a series of concentric shells, or you can look up the answer on Wikipedia:

It's negative because gravity is an attractive force, and this represents the energy that must be overcome to scatter the planet far enough that it won't reform. For the Earth, this comes to 2.2 x 10^33 Joules, so now we need to know how much energy we can get from a star.

The Sun radiates energy at a rate of 3.846 x 10^26 Watts, or Joules/second. If the base collected the light passively, it would take about 66 days to charge up, which isn't ideal. It needs even more energy than that, given that it's able to target multiple planets at once:

Let's suppose it can absorb all the energy the star would produce over it's entire lifetime. For our sun, the estimated lifespan is 10^10 years. It probably won't have the same luminosity over its full life, but I couldn't find a simple model for how it would vary, so we'll suppose it's constant. Multiplying by the power from earlier, this gives 1.21 x 10^44 Joules, or enough to destroy 55 billion planets! Neither situation seems very practical, which may be why the Empire/First Order has so much trouble holding on to power.

Reverse the Polarity

[Title from an (in)famous Doctor Who quote]

 

Last night I went to see Avengers: Endgame in 3D, and I can never resist playing with the glasses afterward, so I thought I'd talk a bit about how they work. In order to get a 3D image, your eyes need to get slightly different pictures – You can see this by alternately closing one eye and then the other. For a long time, this was done with glasses that had one red lens and one blue. The two images would be printed in the same red and blue, so each eye could only see one. Unfortunately, this only works for black & white (or in effect, black & purple) images.

Modern 3D films instead use two different polarizations. I've mentioned polarization before talking about sunglasses, and that's basically all the 3D glasses are, with one important difference: Normal polarized sunglasses are linearly polarized, while the kind used for 3D are circularly polarized. You can think of polarization as an arrow pointing perpendicular to the direction the light is traveling. For linear polarization, this arrow is fixed, but for circular polarization it rotates. The polarization in the linear case is usually described as horizontal and vertical, while for circular it is right- or left-handed*.

We need two different images to get the 3D effect, so we could do that with one horizontal filter, and one vertical. The problem is, if you tilt your head, the two images will mix. To avoid that, films use circularly polarized light, with the glasses constructed to filter right- or left-handed polarizations. To make that filtering happen, the light is first transformed into linear polarization, then filtered leading to some interesting effects. Next time you see a 3D movie, I highly recommend playing with the lenses a bit:

Both lenses forward, 90° rotation

One lens reversed, 90° rotation

*In case you're wondering what physicists mean by the handedness of quantities: You're probably used to talking about rotation in terms of clockwise and counter-clockwise. There's a problem with this though, if you imagine a see-through clock. Viewed from the back, the hands appear to be moving counter-clockwise. To remove this ambiguity, physicists take their right hand and curl their fingers in the direction an object is moving. Extending the thumb points in the direction of the rotation vector. In the case of a clock, this points into the wall.

It's Not the Fall That Kills You

This week, I'd like to get back to one of the mainstays of this blog: Looking at physics in popular media.  A recent favorite movie of mine is the 2015 film Tomorrowland, about a secret pocket dimension where the world's greatest scientists are allowed to do their work in peace.  The central message is that an important part of science is hope for the future, that all is not lost.  Helpful thing to remember after my midweek post.

The opening scene is George Clooney's character, Frank, as a young boy inventing a jet pack.  On his first trip to Tomorrowland, he falls off a ledge, but is able to catch his jet pack, put it on, and stop just before hitting the ground.  Every time I've watched this scene, I've been skeptical that he could survive such a rapid acceleration.

I looked around for human g-force limits, and one of the largest values was for a rocket sled, which had an acceleration of 46.2 g.  A rocket sled sits on a rail, and a rocket at the back propels it (image from Wikipedia):

This is a little different from a jet pack, since it has a seat that can cushion some of the acceleration, so I'm not sure how accurate a limit that is.

In the movie, Frank falls for about 65 seconds before turning on the jet pack.  On Earth, it only takes a person 15 seconds to reach 99% of their terminal velocity, 54 m/s.  Tomorrowland looks to have similar gravity and atmosphere, so Frank is certainly going this fast by then.  When he fires the jet pack, he comes to a stop in about 2 seconds, which works out to 2.8 g, nowhere near fatal!

In case you're wondering what would happen without air-resistance and terminal velocity, even that isn't necessarily fatal: 32.5 g wouldn't be fun, but he would probably get away with only some dislocated shoulders.

On the Rack

Lately I've been watching the anime series One Piece, about a man named Monkey D. Luffy who wants to be the Pirate King.  In the story's world, there is something called Devil's Fruit, which gives those who eat it different supernatural abilities.  In Luffy's case, it made him a rubberman, giving him the ability to stretch his body as if it were made of rubber.  I won't get into the details, but in the episodes I've recently watched, Luffy ended up with one arm encased in a giant gold sphere, leading to this scene:

Since it's a little hard to see out of context, Luffy is holding onto a beanstalk with one arm, while the one with the gold dangles below.  I decided it was the perfect opportunity to see how stretchy Luffy is.

First, I needed to figure out the scale in the picture above.  The white line shows Luffy's height, which I assumed to be the average male height in Japan.  Using this, I found the distance his arms stretched (red) and the diameter of the sphere (blue).  From this, we can calculate that the sphere has a volume of about 1500 cubic meters.  The density of gold is about 19 grams per cubic centimeter, so the sphere weighs 29 million kilograms.

Supposing Luffy's arms follow Hooke's Law (the basic spring force), we can use the weight to find the spring constant.  Setting the weight of the sphere equal to the Hooke's Law force gives a spring constant of 10 million Newtons per meter.  For comparison, the spring constant for the front suspension of a Triumph sports car is about 35,000 Newtons per meter.

Of course, using Hooke's Law is an approximation, since rubber has some unusual stretching properties.  The polymers that make up rubber are best modeled with statistical mechanics, where temperature and entropy play significant roles.  If you're interested, this is a nice explanation.

Zip Line and Sinker

Earlier today, Steve and I went to see Captain America.  It was very entertaining, but as usual, I have a physics nitpick.  In one scene, the Captain and his men ride a zip line down from a mountaintop and drop off onto a speeding train, landing with ease.  It seems to me it would require an enormous height difference to achieve the necessary speed to land on the train without being thrown off.

We can find the height necessary to achieve a certain speed using energy.  The kinetic energy gained by dropping a height h is

where m is the object's mass and g is the acceleration due to gravity.  Meanwhile, the energy involved in traveling a velocity v is

Putting these together and solving for h gives

Given the era, I'm guessing the train was a diesel engine, so its top speed would be around 100 km/h.  Plugging that into our equation gives 39 meters, or about 13 stories.  Also note that this is the minimum height, since it assumes that all his downward momentum gets transferred to horizontal.  If the zip line were at a 45° angle, it would require twice the height.  Don't let my pedantry put you off though; it was a great movie, well worth seeing.

Magneto Gets an F

Last night I watched X-Men: First Class, and enjoyed it very much; it was far better than I remember the earlier movies being.  However, there's something that has always bothered me about the X-Men franchise: it seems Magneto, the mutant who can create powerful magnetic fields at will, never took physics.  There's much more he could be doing with his power, aside from tossing metal things around.

Before I get into that, I'd like to figure out exactly how powerful his magnetic fields are.  During one scene, he lifts a submarine out of the water and into the air:

A little research suggests that, although this is not a real submarine model, the Type XXI is a good approximation.  Using the length of the sub as a scale, we can estimate that the center of the sub (where we'll assume its center of mass is) is about 21 meters above the bottom of the picture.  The energy required to lift something to a specific height is given by

where m is the mass of the object, g is the acceleration due to gravity, and h is the height to which it is raised.  We can get an underestimate of this energy by assuming the submarine was on the surface of the water when Magneto started lifting it.  Plugging in the values we have, it would take 3.3 x 10^8 Joules.  I estimate it takes him about 20 seconds to get it there, giving a power output of 1.7 x 10^7 Watts.  [Update: I realized that, if that energy is coming from Magneto himself, he must be eating at least 80,000 Calories per day.]

With that in hand, we can turn to other uses of Magneto's power.  Maxwell's Equations tell us that changing magnetic fields induce electric fields, the simplest application of that being to create currents in conductors.  However, if he can create 'accelerating' magnetic fields (ones with non-zero second derivative), he can shoot lasers.  Light is made up of oscillating electric and magnetic fields, and since each induces the other, you only need one to get things started.  If he could manage to produce gamma rays (a dangerously high-energy variety of photons), according to our power output calculated above, he'd be putting out 8.6 x 10^20 photons/second.  That's about on the order of how many photons a lightbulb produces, but we're talking about a lightbulb putting out lethal radiation.

I suppose it wouldn't make for a very interesting story if Magneto simply gave everyone on Earth radiation poisoning, so maybe the writers are justified in ignoring this possibility.  It still would have been better than this possibility.

You Can't Fight in Here, This is the War Room!

Yesterday I watched one of my favorite movies, Dr. Strangelove.  During the scene where General Jack D. Ripper pulls a machine gun out of his golf bag and starts firing out his office window, I started wondering how much the barrel would heat up, and whether he'd really be able to hold it with his bare hands.  I remember a scene in some classic war novel (possibly All Quiet on the Western Front) where a group of soldiers cool their machine gun by pouring urine on it, so it seems like a pretty serious concern.

I'm not sure whether the heat comes more from the exploding gunpowder, or from the friction of the bullet against the barrel, but I don't know anything about the chemistry involved in gunpowder, so I'll stick to the friction.  The force of friction is given by

where μ is the coefficient of friction, and N is the normal force on the object being considered.  The coefficient of friction for steel against moving steel is 0.42.  The normal force will be determined by how hard the bullet is pushed against the barrel.  For simplicity, we'll assume it's just gravity causing the contact, although there are probably more complicated forces contributing too, like the rifling of the barrel.  In this case, the normal force is

where m is the mass of the bullet.

To get the total energy deposited by the friction, we integrate over the length of the barrel:

We can turn this into a change in temperature using the specific heat capacity, c:

where M is the mass of the body being heated, the barrel in this case.  Plugging in the numbers for an M2 Browning gives an energy of 0.21 Joules, and a temperature change of 4.6 x 10^-5 Kelvin.  This is a pretty tiny amount, but remember that this is only for a single bullet.  The gun fires approximately 10 bullets every second, so its temperature goes up by 4.6 x 10^-4 Kelvin per second of firing.  I guess that's still pretty tiny; it would take 20 minutes of firing to raise the temperature by 1°F.  Sounds like either most of the heat comes from the gunpowder, or there's much more friction than I estimated.

Hammertime

Last Friday, Steve and I went to see the movie Thor.  It was a lot of fun, but Steve keeps insisting the movie was about physics, so I figured I should do a post on it.

One of Thor's powers is that he can use his hammer, Mjöllnir, to fly.  He does this by essentially throwing the hammer, but not letting go, so that it carries him along with it.  My first instinct was that this wouldn't work, but I figured I should go through the mechanics, just to be sure.  Let's say he holds the hammer at his side, and swings it up in a circular arc until it's above his head.  He'll be able to accelerate it until it's straight in front of him, but after this point it will begin to lift him.  Lets say at this point it's moving at a velocity v.  If the hammer has mass m and Thor's arm is of length r, then the force required to maintain it's circular motion is

Once the hammer reaches this point, Thor won't be able to add more energy to it, so let's look at what happens after this.  The lifting force from the hammer will be

where θ is the angle of Thor's arm above horizontal.  If Thor's mass is M, then his upward acceleration will be

θ is a function of time, so we can put it in terms of other variables we have.

Integrating this over the arc will give the velocity of Thor when he finishes the swing.

or a kinetic energy of

That means he could rise to a height of

To get an idea of what this means, let's get some numbers. We'll assume Thor has an average weight of 70 kg.  A quick search for sledgehammers shows a typical weight is about 7 kg.  It's a little tougher to figure out the speed.  I searched around a bit for info, but the best I came up with was this video of a guy swinging a club of similar weight.  He manages about 1 round per second, which translates to a velocity of about 6 m/s.  Thor's a god, so let's estimate he can manage more than double this, and call it 15 m/s.  Plugging all this in gives h = 11.5 cm.  Of course, if we assume Thor has truly god-like strength, then he can swing Mjöllnir as fast as he wants, or we could assume Mjöllnir is stunningly heavy.  Either way, he could manage flight, but it's certainly outside the abilities of any human.

There was one physics detail that really bugged me in the movie.  For much of the film, Mjöllnir is wedged in the ground out in some desert, with a government research team surrounding it.  Occasionally, the team's computers flicker, which they explain is the result of powerful electromagnetic energy being released by the hammer.  What bugs me is that none of them think to simply erect a Faraday cage around it.  Faraday cages absorb electromagnetic waves, and are often used in labs to protect equipment from volatile experiments.  Swarthmore's plasma physics lab uses one to protect their computers from the powerful waves emitted by their spheromak.  You may think I'm being petty, but I feel strongly that if a movie is actually going to talk about science, they should get it right.