Name: James Flaten, Alex Nugent, Andrew Foltz

Institution: Luther College

Address: 700 College Drive
Decorah, IA 52101

Phone: 563-387-1632

Fax:

E-mail: flatja01@luther.edu

Apparatus Title: Illustrating Wave Phenomena with Overhead Projection of Sets of Pendula

Abstract (50-75 words)

Sets of identical pendula, suspended above an overhead projector, can be used to demonstrate the oscillations of individual particles in a line during wave phenomena. Pendula double-slung so as to swing perpendicular to a line of bobs can be used to show transverse traveling waves, free-end reflections, and transverse standing waves. Pendula can also be double-slung so as to swing along a line of bobs, to demonstrate particle motion in longitudinal wave situations.

Equipment and costs required to construct apparatus:

Item	Source	Part number	Cost
3/8” steel hunting shot (75 pieces)	Walmart	Marksman 3138	$2.96, enough for 5 sets of 15 pendula each
button and craft thread (75 yards)	Walmart	e.g. Coats CA00011	$0.96, enough to suspend many sets
super glue	hardware store	e.g. QuickTite brand	$2.99
1” wood screws (box of 100; need 16 per frame)	hardware store		$2.49, enough for many frames
0.5” x 0.75” wood (AKA parting stop) ~12 feet per frame	lumber yard		$4.80 for 16’, enough to build one frame
thin wood meter stick	hardware store		$0.99, each stick hangs 2 sets of pendula
Total Cost			$30.58 to build 4 different sets – plans for 2 transverse sets shown below

Description:

Sets of identical pendula, suspended so as to form a 1-dimensional line of oscillators, can be used to demonstrate various aspects of pulse and wave motion that are difficult to show with continuous media like coil springs. Shadows of pendulum bobs can be projected onto a screen using an overhead projector, allowing an entire class to have a good vantage point from which to observe the motion. This projection scheme also minimizes the size, and hence the cost, of the apparatus. Pendula are double-slung to restrict their motion to either transverse or longitudinal, with respect to the line of pendulum bobs. If the pendula are coupled, pulses can be propagated along the line, free-end (or fixed end) reflections can be illustrated, and standing waves can be generated. If the pendula are uncoupled and released appropriately (as described below), traveling wave patterns with no end reflections can be demonstrated.

Figure 1 below shows a set of identical double-slung pendulum bobs that are uncoupled, so they may swing independently in a transverse fashion. Consider holding all the pendula off-center by the same amount using a rod (not shown) running parallel to the set of bobs. If the rod is removed by pulling it along its length, the bobs will be released at regular intervals and a sinusoidal wave shape will result, moving in the direction the rod was extracted. Since the pendula are uncoupled, this shape is not actually passed from one oscillator to the next, so there are no reflections at the ends. Hence the pendula will exhibit a traveling wave shape (though not an actual traveling wave in a technical sense), something that is difficult to do using a standard, finite apparatus. The transverse motion of the individual particles as the wave moves past is easy to see using this apparatus.

Figure 2 below shows a set of identical double-slung pendulum bobs that are hung from a mesh of strings and hence are (weakly) coupled. The threads are super-glued or tied together at crossing points, to ensure uniform coupling. Again the bobs are restricted to swing in a transverse fashion. With these coupled pendula a transverse pulse started at one end of the set will travel down to the far end then reflect off the free end and return on the same side as the original pulse. If one end of this pendulum set is driven back and forth (by hand) at the appropriate frequencies, transverse standing waves can be generated, with motion nodes and antinodes clearly visible. One main advantage of this demonstration is the fact that the overall wave motion, plus that of the individual oscillators, is slow enough to follow by eye.

Figure 3 below shows how pendulum sets can be suspended inside a wooden frame that may then be placed on an overhead projector. The wooden piece to which the strings are tied must be fairly narrow so that it does not optically obscure the balls below it. As long as this is the case, it is easy to focus the projector on the balls. The connections between the top and bottom squares are not made at the corners, to give better access to the end balls in the set. The balls run diagonally across the projector so that more balls can be put in one set. About 14 or 15 balls is sufficient to illustrate most wave phenomena. Heavy (metal) balls are preferable to lighter (wood or plastic) balls to combat twist in the string/thread used to suspend the balls. Stainless steel bobs may need to be roughed up a bit when super-gluing the strings to the balls.

Sketch(es) (computer generated if possible):

Figure 1 – Uncoupled double-slung pendula to illustrate transverse traveling wave patterns with no reflections that might produce standing waves. Pendula are spaced every 0.75 inches, hanging from a thin wooden yard stick. Length of pendula is about 8 inches.

Figure 2 – Weakly coupled double-slung pendula to illustrate propagation of transverse pulses, reflections off free ends, and transverse standing waves. Super-glue or tie the strings together at the intersection points. Pendula are spaced every 0.75 inches, hanging from a thin wooden yard stick. Length of pendula is about 8 inches.

Figure 3 – Open wooden frame to hold pendula sets above overhead projector. Opening is 10 x 10 inches (size of overhead projector surface) and height is about 9 inches (must fit under overhead projector upper lens).