Physics Quanta: Pendulums Revisited

(Note: this is part 2 of a new series of posts, in which I explain a basic physics concept and put it in a wider context. Since the title of the blog is Galileo’s Pendulum, I thought starting with the pendulum would be appropriate. A quantum is a small quantity of something; the plural is quanta, so the series “Physics Quanta” consists of bite-size chunks of physics.)

Last week, I began with the simple pendulum, so go back and reread that post before starting this one. (I’ll wait here; no rush.) Two things we assumed last week: that the pendulum swings through a small angle, and that air resistance is negligible. Let’s relax each of these assumptions in turn.

Pendulums Swinging Through Any Angle

If the bob is attached with a wire instead of a string, there is a second equilibrium position at the top of the swing

To illustrate a pendulum swinging through a large arc (but still ignoring the friction from air), it’s best to think of the bob attached to a metal rod rather than a string. The reason for this is simple: just like you can sometimes balance a stick or pencil on its end, the pendulum can balance precariously at the absolute top of its swing, 180º from the other equilibrium position. The slightest nudge will bring the pendulum swinging down, so the top point is called unstable equilibrium. (If you think about it, a human being standing is in a kind of unstable equilibrium. We achieve balance by the active exertion of our muscles.)

You can start the pendulum swinging in three ways: lift the bob up to some angle and let it go, or start it at its lowest point and give it a push, or start it at some other angle with a push. In other words, there are two different physical quantities that dictate how the pendulum swings: its initial angle and its initial speed. In fact, if you give the pendulum a hard enough shove, it can rotate in a full circle.

To figure all this stuff out, physicists use a type of plot known as a phase diagram. (The “phase” here is not to be confused with phases of matter, like the transition between liquid water and ice, which are also mapped on something called a phase diagram. Physicists can get confusing sometimes.) In a phase diagram, the angle of swing is plotted on the horizontal axis and the velocity of swing is plotted on the vertical axis; negative numbers just mean the position is on the left of equilibrium or the bob is moving to the left.

Dark blue and green lines: small speeds, so the pendulum just swings back and forth. Red: larger speeds, so the bob comes close to the unstable equilibrium point. Cyan: enough speed so that the pendulum swings in full circles

What we find is that the pendulum traces definite shapes in the phase diagram. The plot on the right shows the same pendulum starting at the same angle, but with four different amounts of push. Each dot represents a different point in time, so we’re seeing a kind of stroboscopic picture, only with the extra bit of information from the velocity. The four trajectories are as follows:

  1. The dark blue circle is what happens if we just release the bob from rest. It traces a small circle, and will keep tracing that circle forever in the absence of air resistance. The period of oscillation (which I wrote about in the previous post) is the length of time for the trajectory to repeat itself.
  2. To get the green circle, we give the pendulum a bit of a push, but not too much, so it never swings too far up.
  3. For the red dots, we give it a big push, but not quite enough for it to reach the unstable equilibrium point at the top of the swing. However, it slows waaaaay down when it gets near the top, which flattens the circles we saw in the blue and green trajectories into something more eye shaped.
  4. The cyan trajectory is if we give the pendulum a big shove, enough that it will keep turning in circles. However, it’s not going to swing at a steady speed, which is why the line isn’t straight. Close to the unstable equilibrium position, it will be moving the most slowly, while it will be moving fastest near the stable equilibrium position at the bottom of the swing.

I also included yellow arrows that indicate what a pendulum having that particular position and speed will do in the future. The longer the arrow, the more quickly it will transition to its new point in the phase diagram. There is a special point right in the middle where the position and the velocity are both zero: this represents the pendulum hanging straight down with no motion — in other words, at stable equilibrium.

I realize this is probably a new way to visualize motion — typically, the position is plotted as a function of time. In this case, though, because there’s repetition, we can convey more information by sort of taking time out of the picture. Each dot is a snapshot in time, but if you look at the whole trajectory, you see that pushing the bob at different speeds produces different types of motion. That kind of thing is hard to see with traditional position-vs.-time plots.

With Air Resistance

The starting positions and velocities are exactly the same as in the previous plot, but this time air resistance is included

Air resistance depends on the an object’s size, shape, and mass. A heavy, compact object will experience relatively little resistance, while a large, light object will experience a lot. Adding air resistance to our pendulum model while leaving everything else the same has a simple effect: all trajectories will end at the stable equilibrium point! Instead of circles, eye shapes, and wiggly lines, we get spirals. The amount of air resistance is pretty small in this picture, but cranking it up (by using a lighter and/or larger bob, or even by immersing the bob in some liquid) may make any kind of oscillation cease!

Air resistance (along with friction) explains why old-fashioned pendulum clocks need winding up, or some other method to keep them going. Even the best pendulum will experience some air resistance, so the clockwork mechanism helps keep the bob swinging back and forth at a regular rate.

Things can get even more fun if we add a driving force to the pendulum, but I’ll save that for another day. The next installment will connect pendulum oscillations to quantum mechanics and other physical systems that don’t at all look like a mass on the end of a string.

9 responses to “Physics Quanta: Pendulums Revisited”

  1. […] in which I explain a basic physics concept and put it in a wider context. The first two dealt with pendulums; I do have one more post about pendulums to come as […]

  2. […] Physics Quantum introduced the basic physics of ideal pendulums undergoing small oscillations; the second extended the idea to large oscillations and added in air resistance. As I promised, this is the third pendulum-related post, connecting the physics of oscillators to […]

  3. […] series explaining basic physics concepts, starting with simple oscillators (pendulums) and going from there. (I’m not done with oscillators yet, either — I intend to do at least three […]

  4. […] Yesterday (September 18) was the birthday of Jean Bernard Léon Foucault (1819–1868), usually known as Léon Foucault amongst his fans. Foucault is best known for his realization that a pendulum allowed to swing in any direction (instead of back and forth in a single plane of motion) on a sufficiently long chain could show the rotation of Earth. Known as a Foucault pendulum today, these devices are a staple of many science museums, including my local museum, the Science Museum of Virginia. I will write a post about Foucault pendulums soon(ish), as a continuation of the general oscillator series. […]

  5. […] of oscillation dictated by the difference in masses between the flavors. (Have I pointed out yet how many things in nature are oscillators?) Interestingly, we have a better handle on the difference in masses than […]

  6. […] Now for some physics, which I will definitely elaborate in a follow-up post. An electric charge or a mass is akin to a hole in a surface: going to three dimensions instead of two, a closed curve becomes a closed surface. A surface enclosing a charge or mass is not the same as a surface enclosing empty space, and they can’t be deformed into each other without tearing holes. The strength of the electric or gravitational force is then determined by the size of the surface containing the charge or mass. (Those with a bit of physics from high school or college may recognize the origins of Gauss’ law in what I’m writing here.) Farther afield, the central singularity of a black hole is a true topological defect, a place where the normal laws of physics break down. We can even talk about topology in the context of simple pendulum dynamics: in phase space, the points of equilibrium, where a pendulum comes to rest,…. […]

  7. […] position can be plotted on the same axes, but they aren't really perpendicular quantities. (See my earlier post for a complete explanation of this […]

  8. […] its position, taken over many cycles. A normal pendulum would describe a circle (or more accurately a slow spiral inward as air resistance slows the pendulum down), but this is not ordinary. In fact, I constructed this […]

  9. […] more about phase portraits, see my earlier posts on pendulum motion. For a lot more information, I recommend reading up on nonlinear dynamics! See Steven H. […]

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