Circular Motion and SHM
simple harmonic motion - uniform circular motion - periodic motion
What it shows:
One of the simplest of periodic motions is uniform circular motion. By shadow
projecting both uniform circular motion and oscillatory simple harmonic motion
onto a screen, one can show that these two seemingly different kinds of motion
are actually identical.
How it works:
A 8 cm diameter plastic ball mounted near the edge of a 46 cm diameter disk
undergoes uniform circular motion. The disk, oriented vertically, is driven
by a 57 RPM motor.
1
A large iron weight (14.87 kg), suspended from a stiff spring, is the simple
harmonic oscillator. The spring/weight combination has been carefully chosen
so that its period of motion is almost identical to the motor.
frequency ≈ 1 Hz
The oscillating weight and rotating ball are positioned as shown. Not shown is a
projection screen off to the left and a slide projector light source far away on
the opposite side of the lecture hall. The audience view of the rotating disk is
clearly uniform circular motion. The lighting on the disk is edge on, and so the
shadow is that of the ball oscillating vertically. The shadow of the weight of
course looks the same as the front view of it. The weight is raised from its
equilibrium position and released at the appropriate time. The two shadows (of
the weight and ball) oscillate vertically in unison -- a striking demonstration
of the relation between these two very different yet very similar motions.
Setting it up:
The spring/weight combination is suspended from heavy duty lab clamps and rods
(3/4" diam) on a lecture cart. The disk/motor sits on the same cart and the
entire ensemble is positioned close to the projection screen so that the shadows
are nice and sharp. A slide projector (with or without lens) on an AV cart sits
on the opposite side of the lecture hall -- the further away, the sharper the
shadows. Move the carts to adjust the lighting so that the shadow of the
rotating disk is just a vertical line on the screen.
Comments:
Because the frequencies of the two systems are not perfectly identical, the shadows
start to noticeably get out of phase with each other after a dozen or so oscillations.
The demonstration can be stopped before that happens. Rating **
1 Bodine Electric Co type NS1-33R speed reducer motor (1725 RPM with 30:1 gear ratio). The disk is cut from 1/8" thick masonite.