- Published: Monday, 10 August 2020 09:00
Today we’re looking at two demonstrations that are often used, individually or together, to discuss simple harmonic motion. Demonstration G1-11: Comparison of Simple Harmonic Motion and Uniform Circular Motion, is a simple mechanical model with a large rotating arm with a disc mounted on it. As the arm-mounted disc rotates around the center, we can see that its motion describes a circle in space. The arm is linked mechanically to a second disc mounted above, that slides back and forth as the arm rotates. The upper disc keeps pace with the lower disc, and as the arm rotates, the upper disc moves back and forth as though it were mounted on a spring.
Demonstration G1-12: Pendulum and Rotating Ball, lets us see that this is not just a coincidence of the model. A ball is mounted as the bob on a rigid pendulum, while an identical ball is mounted on a rotating platform below. The rotating platform is motorized so that it will spin at a constant speed; the pendulum is of an appropriate length so that the period of the swing is the same as the rotational period of the platform. If you start them moving from the same point at the same time, then you can see that the two balls move in sync. By positioning a bright light in front of the apparatus we can project the shadows of both balls on the wall behind, and we can see that the two balls are executing nearly the same motion.
A ball executing simple harmonic motion – the motion of a pendulum bob – is equivalent to the projection of a ball executing uniform circular motion. This is not just a coincidence of the apparatus, but a fundamental discovery about the mathematics behind repeating motion.
(diagram based on public domain work by Wikimedia user Yohai)
If we make a graph of the linear position of a point on the rotating disc as a function of time, that graph traces out a repeating curve – a curve we can describe with the cosine function, Acos(ɷt), where A is the radius from the center of the circle to the point and t is time. For those of you who have studied the behaviour of harmonic oscillators, that function should look familiar – it’s the same way we describe an object oscillating without damping, what’s called simple harmonic motion. ɷ (omega) is the rate of rotation of the disc, and equivalent to the angular frequency of the oscillation. And conversely, if you made a graph of the velocity of an oscillating mass against its position, rather than plotting the position or the velocity against time, that graph would also trace out a circle. It’s not just a coincidence, but reality – rotational motion and oscillating motion are fundamentally the same phenomenon from a mathematical perspective, just looked at in different dimensions.
(PD Animation credits: Wikimedia users Chetvorno & Mazemaster)
This simulator, by Andrew Duffy of Boston University, lets us model this behaviour on the screen, and see what happens when we change parameters of the motion. Check it out at http://physics.bu.edu/~duffy/HTML5/SHM_circular_motion.html .
This simulator lets us view this motion in real time. Press Play and see a point rotating on the disc, while two more masses oscillate on springs vertically and horizontally next to the disc. The graph plots out the vertical motion of both the point on the disc and the vertical oscillator over time. You can click the checkbox at the bottom of the screen to form virtual lines between the masses, to show they’re in sync.
Now try changing the experiment. There are two sliders at the bottom of the simulation. The slider on your left lets you change ɷ – try speeding it up and watch what happens! The slider on your right lets you change the radius of the disc, and thus the amplitude of the oscillation.
Try it out for yourself! And think about where else you’ve seen graphs like that. There are many other physical phenomena that obey similar mathematics, including all types of waves. What examples can you think of?