Friday, May 31, 2019

Metrological Forecast

Last week, the International Committee for Weights and Measures (BIPM) definition of the kilogram changed. Since I'm in the home of the former kilogram model, Le Grand K, I thought I'd take the opportunity to talk about Metrology, the process of measuring quantities for study.

Many of the units in Physics are defined in terms of others. Energy is often measured in Joules, but these are equivalent to kilogram meters squared per second squared. You still need a base set of units for a few fundamental quantities to define the others. The outer ring of wheel above shows the quantities in the International System of Units (SI) that are used as a basis for all others:
  • mass (kilogram)
  • distance (meter)
  • time (second)
  • current (Ampere)
  • temperature (Kelvin)
  • number of particles (mole)
  • luminous intensity (candela)
Using these 7 quantities, all other units can be defined, but the problem remains of how to define these 7 precisely. To ensure consistency across the planet (and indeed, the universe) these 7 units are each derived from a measurable universal constant, shown in the inner circle.

For example, the meter is defined in terms of the speed of light, c: 1 meter is the distance light travels in 1/299,792,458 second. The second, in turn, is defined as 9,192,631,770 oscillations of a caesium atom. These unwieldy numbers come from the fact that the units existed before they were formally defined in terms of constants: Most people would say a second is 1/(60 * 60 * 24) of a day, but a day is not a precise enough quantity for certain measurements.

That brings us to the kilogram: Until last week, mass was defined in terms of an actual bar of platinum and iridium referred to as Le Grand K:
Reuters/Benoit Tessier
The problem with this method is that any change to that lump of metal effectively changes what a kilogram is. Comparing the base model to duplicates in 1989 showed that the duplicates were heavier. It's impossible to know though whether the duplicates had gained weight since their creation, or Le Grand K had lost weight, since no matter what, it weighed 1 kg. The redefinition resolves that problem.

The new definition uses Planck's constant, h, to set the kilogram. This is a quantity used in quantum mechanics, which relates the frequency of light to the energy it carries. It was first used to describe blackbody radiation, the light emitted by heated objects like the sun, or even an incandescent bulb.  Since h has units of energy * time, we can use the other definitions to find the kilogram. Here's a nice diagram similar to the one at the top, but showing how the units interconnect:
One of the things I love about science is how we have a good understanding of how things work now, but we still strive for greater precision, which in turn reveals new areas to explore. The universe is a beautiful place, and I'd like to appreciate every second, meter, kilogram, mole, candela, Kelvin, and Ampere of it!

Saturday, May 25, 2019

Wheels Within Wheels

At the beginning of this week, my parents returned home from their 40th anniversary trip around France. As a souvenir, Sally asked me to pick up one of the great wheels of regional Tomme cheese from the Annecy farmers' market. While delivering it to their hotel, feeling the weight of it got me thinking about timekeeping of all things.

I imagined letting the cheese roll in a large dish, with each crossing representing one "tick". This is similar to how a pendulum clock works, but with an important difference: Rolling is subject to rotational inertia. I was curious how that would change the frequency, but before I get to that, I should explain what rotational inertia is.

Inertia refers to an object's tendency to keep moving in the way it's currently moving. Changing an object's motion requires overcoming its inertia. For linear motion, the inertia is simply the mass: It's a lot easier to catch a baseball than a bowling ball. Rotational inertia is the same idea, but the movement is around a central axis. A top stays upright because its rotational inertia keeps the spinning in the same direction.

Rotational inertia depends on the mass, but also how far that mass is from the rotational axis. You can look here for the inertias of some common shapes, but they generally follow the same form:
where q is a factor between 0 and 1. For the animation below, I chose a hollow ring, a solid disk (like the cheese), a solid sphere, and finally, a frictionless object that only slides without rolling. These have q = 1, 1/2, 2/5, and 0 respectively.

Each of these objects moves inside a larger circular container. The position in this container can be defined by the angle off the center. Aside from the frictionless object, the others roll without slipping, meaning we can relate their rotation to the position in the larger ring:
where L is the arc-length from the bottom of the ring to the object, and the rs and θs correspond to the radii and angles of the large ring and the object.

I'll spare you the details, but the acceleration for these objects winds up being
and so the angular position is
What this says is that as q gets bigger, the frequency (equal to the factor multiplying t) decreases. Looking below, we see the objects with larger q do indeed lag behind the others.

Sunday, May 19, 2019

Intelligent Falling

[Title refers to the equally credible explanation for gravity.]

I'm in the middle of reading Seveneves, by Neal Stephenson, which involves humanity fleeing the surface of Earth to survive in orbit. All the discussion of orbital mechanics reminded me of a program I loved playing with when I was growing up called Gravitation, Ltd. It let you set up bodies of different masses in space, and let them move according to each other's gravity. What I particularly liked seeing was the different patterns the orbits would make. Since that application is more than 20 years old at this point, I was curious if I could set up another HTML5 widget to play with.

For any pair of objects, Newton's Law gives the acceleration of one object created by the other:
where G is Newton's constant, m is the mass of the attracting object, and r is the vector from the accelerating object to the attracting one. This is a second-order differential equation for the position of the body: The equation gives an acceleration, which is the change in velocity, which is the change in position. At every step of time in our simulation, we need to find the acceleration, and from that get the velocity and position. The natural way to do that, integrating up from acceleration, would be
where the subscript indicates the step number. This is called Forward Euler Integration. For certain systems though, this method can be unstable, producing results that diverge to infinity. A more robust technique is Backward Euler Integration, which does the calculation in the opposite order:
Both are implemented below, but I haven't noticed much difference for the cases I've tried. Play around a bit, and be sure to post interesting settings in the comments!

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Initial Position

Initial Velocity

Star Mass


Saturday, May 11, 2019

From a Certain Point of View

[Title from Obi-Wan's BS explanation for lying to Luke]

When I first put up the post about my stair-climbing cart, my father Steve was disappointed there was no video of it in action. At the time, I had hoped to create an animation of the movement, but I couldn't get my head around the equations involved. This week, I had an idea though: The system is fairly simple to describe if we only consider the cart rotating around one of its wheels. Bringing back the diagram from last time,
we can imagine holding the front wheel fixed, and letting the other two rotate around it.

In Physics, this is called choosing the reference frame. Essentially, we're deciding where the x- and y-axes are, and what we're calling zero. Reference frames can also move relative to each other. As long as that velocity difference is constant, all the same physical laws apply. In accelerating frames, things can get a bit crazy, but we don't have to worry about that here.

In this case, we can put the corner of the front step at (0, 0). We can also rotate 45° to make the stairs horizontal, which makes the motion of the cart symmetric. Here's an animation of what the system looks like going through one step:
Now that we have that simpler motion defined, we can apply a couple transformations to bring it back to the view we're interested in: We can rotate back 45° to line up the stairs, and then follow the cart as it moves, going up one step for every turn.
If you'd like to see the equations that define this, you can dig into the code.

Sunday, May 5, 2019

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.