physics

Time for an experiment! Find a book and secure it shut using tape or a rubber band. Now experiment with spinning the book while tossing it into the air. You’ll notice that when the book is spun about its longest or shortest axis it rotates stably, but when spun about its intermediate-length axis it quickly wobbles out of control. Every rigid body has three special, or principal axes about which it can rotate. For a rectangular prism — like the book in our experiment — the principal axes run parallel to the shortest, intermediate-length, and longest edges, each going through the prism’s center of mass. These axes have the highest, intermediate, and lowest moments of inertia, respectively. When the book is tossed into the air and spun, either about its shortest or longest principal axis, it continues to rotate about that axis forever (or until it hits the floor). For these axes, this indefinite, stable rotation occurs even when the axis of rotation is slightly perturbed. When spun about its intermediate principal axis, though, the book also continues to rotate about that axis indefinitely, but only if the axis of rotation is exactly in the same direction as the intermediate principal axis. In this case, even the slightest perturbation causes the book to wobble out of control. The first simulation above shows a rotation about the unstable intermediate axis, where a slight perturbation causes the book to wobble out of control. The second and third simulations show rotations about the two stable axes. Unfortunately, as far as my understanding goes, there’s no intuitive, non-mathematical explanation as to why rotations about the intermediate principal axis are unstable. If you’re interested, you can find the stability analysis here. Mathematica code posted here. Additional sources not linked above: [1] [2] [3] [4] ...

Gabriel’s Horn is a three-dimensional horn shape with the counterintuitive property of having a finite volume but an infinite surface area. This fact results in the Painter’s Paradox — A painter could fill the horn with a finite quantity of paint, “and yet that paint would not be sufficient to coat [the horn’s] inner surface” [1]. If the horn’s bell had, for example, a 6-inch radius, we’d only need about a half gallon of paint to fill the horn all the way up. Even though this half gallon is enough to entirely fill the horn, it’s not enough to even coat a fraction of the inner wall! The mathematical explanation is a bit confusing if you haven’t taken a first course in calculus, but if you’re interested, you can check it out here. Mathematica code: x[u_, v_] := u y[u_, v_] := Cos[v]/u z[u_, v_] := Sin[v]/u Manipulate[ParametricPlot3D[{{x[u, v], y[u, v], z[u, v]}}, {u, 1, umax}, {v, 0, 2*Pi}, PlotRange -> {{0, 20}, {-1, 1}, {-1, 1}}, Mesh -> {Floor[umax], 20}, Axes -> False, Boxed -> False], {{umax, 20}, 1.1, 20}] Additional source not linked above. ...

The Lagrangian points are the five locations in an orbital system where the combined gravitational force of two large masses is exactly canceled out by the centrifugal force arising from the rotating reference frame. At these five points, the net force on a third body (of negligible mass) is 0, allowing the third object to be completely stationary relative to the two other masses. That is, when placed at any of these points, the third body stays perfectly still in the rotating frame. The first image shows the fields due to the first mass, the second mass, and the rotating reference frame. When added together, these fields generate the effective field shown in the second image. The five Lagrangian points are indicated with gray spheres. The first three Lagrangian points (labeled L1, L2, and L3) lie in line with the two larger bodies and are considered metastable equilibria. L4 and L5 lie 60° ahead of and behind the second body in its orbit and are considered stable equilibria. Lagrangian points offer unique advantages for space research, and the Lagrangian points of the Sun-Earth system are currently home to four different satellites. Mathematica code posted here. Additional sources not linked above: [1] [2] [3] [4] [5] ...

The Doppler effect is the shift in the frequency of a wave observed when the source of the wave (or the medium through which the wave travels) is moving relative to the observer. We’re most familiar with the Doppler effect as it appears in sound waves traveling through air (i.e., pressure waves) – think of how the pitch of a siren drops as an emergency vehicle passes you. The first GIF shows a stationary source and the second shows a source moving to the right at 40% the speed of sound. Notice in the second GIF how the wavefronts are closer together in front of the source (producing a higher frequency) and further apart behind it (producing a lower frequency). The Doppler effect is interesting in its own right, but things get much more exciting when the source travels at speeds greater than or equal to the speed of sound. When the source travels at the speed of sound (GIF 3) the source will always be at the leading edge of the waves it produces, and when traveling faster than the speed of sound (GIF 4), the source will always be in front of the waves it produces. In both of these cases, notice how the waves overlap with each other. The high pressure areas of each wave constructively interfere and produce a region of extremely high pressure (much higher than in the surrounding areas). This rapid rise in air pressure is a shock wave, and the sound associated with it is a sonic boom. In each of the GIFs above we see the radiating wavefronts on the left, and the pressure distribution and interference of the waves on the right. Mathematica code posted here. Additional source not linked above. ...

A chaotic system is one in which infinitesimal differences in the starting conditions lead to drastically different results as the system evolves. Summarized by mathematician Edward Lorenz, “Chaos [is] when the present determines the future, but the approximate present does not approximately determine the future.” There’s an important distinction to make between a chaotic system and a random system. Given the starting conditions, a chaotic system is entirely deterministic. A random system, on the other hand, is entirely non-deterministic, even when the starting conditions are known. That is, with enough information, the evolution of a chaotic system is entirely predictable, but in a random system there’s no amount of information that would be enough to predict the system’s evolution. The simulations above show two slightly different initial conditions for a double pendulum — an example of a chaotic system. In the left animation both pendulums begin horizontally, and in the right animation the red pendulum begins horizontally and the blue is rotated by 0.1 radians (≈ 5.73°) above the positive x-axis. In both simulations, all of the pendulums begin from rest. For more information on how to solve for the motion of a double pendulum, check out my video here. Mathematica code posted here. ...