## Coupled Pendulums

December 2, 2011

by:  Martin Sagendorf

One Pendulum…

Is interesting, but…

Two Pendulums…

Are much more interesting.

But Only If…

They are coupled together.

An Easy Way Is To…

Couple them at their pivot points.  This is accomplished by hanging the two pendulums from a horizontal string.

There Are…

Many illustrations of coupled pendulums on the web; search for ‘coupled pendulums’ – but the fine points of making a really successful demo are rarely discussed… so before we start:

Some Guidelines:

-       Make the pendulums absolutely identical: both the rod lengths and the mass values (the lengths are measured from the pivot points to the C.G. of the masses)

-       Use rod lengths of at least 1/3 meter (13”) – so the pendulums don’t swing too quickly

-       Use masses of at least 75 g (1 oz) – to provide a long swing time

-       Space the vertical supports for a horizontal string length of 500 to 600 mm (20 to 24 in.) – weighted or clamped-down ring stands will work – and will work especially well if their top ends are joined by a solid bar to minimize vibrations

-       The string should be fairly taunt – for example:  a 13 to 15 mm (1/2 to 5/8 in.) droop in the center with two 75 g masses hanging 100 mm (4 in.) apart

-       Use pendulum spacings of 75 to 125 mm (3 to 5 in.) – experiment for good results

-       For the best results, symmetrical setup spacing is critical – try to achieve positions symmetric within 4 mm (1/8 in.)

-       When pulling a pendulum to the side, two things are very important: first, don’t pull it too far (a mass rise of 75 mm (3 in.) is fine); second, the pendulum must be pulled at precisely a right-angle to the string

-       For the following exercises, when two pendulums are raised, they should be raised to the same heights

With Two Identical Pendulums:

Center the two pendulums with the pair spaced about 100 mm (4 in.) apart

-       (A.)  Raise and release one pendulum

Question:  What happens?  Why?

-       (B.)  Raise (on opposite sides) and release both pendulums

Question:  What happens?  Why?

With Three Identical Pendulums:

Center the three with a space of about 75 mm (3 in.) between each

-       (C.)  Raise and release the center pendulum

Question:  What happens?  Why?

-       (D.)  Raise and release one of the outer pendulums

Question:  What happens?  Why?

-       (E.)  Raise (on the same side) and release both outer pendulums

Question:  What happens?  Why?

-       (F.)  Raise (on opposite sides) and release both outer pendulums

Question:  What happens?  Why?

So Far…

We have dealt with identical pendulums… but what happens if we:

-       (G.)  Make a pendulum with a greater mass (but the same length) and use it in place of one of those

above

Question:  What happens?  Why?

-       (H.)  Make a pendulum just slightly longer (say, 20%) than one of the three and use it in place of one of

the pendulums above

Questions:  What happens?  Why?

In Action:

Construction Notes:

-       The horizontal string must be firmly attached (tied, hooked, or taped) to the vertical rods

-       The pendulum rods are made from coat hanger wire or from welding rod

-       Hooks are formed in the pendulum rods using a pair of pliers

-       The masses can be any object that can be affixed to the rod – preferably an object through which a hole can be drilled and, for easy identification during demonstrations, the masses should be different colors

In This Apparatus:

-       Length of horizontal string = 600 mm (23-1/2”)

-       Length of pendulum rods (from inside hook to far end) = 440 mm (17-7/16”)

-       Diameter and material of pendulum rods = 1/8” brass welding rod

-       Thread on end of pendulum rod = 6-32 for a length of ¾ in. (Note 1)

-       Nuts = brass 6-32 knurled (2 per rod)

-       Small mass = 5/8” x 2-1/16” steel rod (75 g) – 3 required (Note 2)

-       Large mass = 1” x 1-3/4” steel rod (175 g) – 1 required (Note 2)

-       Distance from inside of pendulum rod hooks to the centers of masses = 400 mm (15-7/8”)

Note 1:  A No. 6 screw diameter is 0.138”. – the 1/8 in. welding rod is 0.013” less – this is OK

Note 2:  Drilled thru No. 29 (0.136”)

A Comment on Dimensions:

The overall dimensions are not critical, but the apparatus should be large enough to be easily viewed in a classroom setting.

A Definition:

These are ‘Simple Pendulums’ because they are not ‘ideal’: i.e. their masses are not concentrated at single points and the restoring force is not a constant – however they do exhibit ‘Simple Harmonic Motion’.  This motion is an approximation at small angles – it is sufficiently accurate for our purposes.

And Further:

The details of Harmonic Motion and Simple Harmonic Motion are fascinating – the details of both can be found in any physics textbook.

‘Resonance’ is defined as the building up of large vibrations by the repeated application of small impulses whose frequency equals one of the natural frequencies of the body – in this case, a pendulum.  Identical pendulums are required to provide maximum energy transfer.  The mechanical energy is transferred by the ‘pulls’ on the supporting string – this is rather like a child’s swing where ‘pushes’ applied at the correct times will ‘add’ and act to increase the swing amplitude.

In Summary:

These demonstrations provide vivid illustrations of energy transfer between two and three resonant bodies.  Even better, additional pendulums, various masses, and variations of excitation will provide more interesting demonstrations and bases for experimentation.

Marty Sagendorf is a retired physicist and teacher; he is a firm believer in the value of hands-on experiences when learning physics.  He authored the book Physics Demonstration Apparatus.  This amazing book is available from Educational Innovations, Inc. – it includes ideas and construction details for the creation and use of a wide spectrum of awe-inspiring physics demonstrations and laboratory equipment.  Included are 49 detailed sections describing hands-on apparatus illustrating mechanical, electrical, acoustical, thermal, optical, gravitational, and magnetic topics.  This book also includes sections on tips and hints, materials sources, and reproducible labels.

May 21, 2010

by:  Martin Sagendorf

An Odd Name: They’re named for the German physicist Ernest Chladni who popularized them in the mid-1700s.  His name is pronounced: kläd’nêz.

They are: Thin plates (sprinkled with fine particles) vibrated perpendicular to their plane.

How? – Then and Now: Long ago Chladni used a cello bow to excite the edge of a thin metal or wooden plate.  Today, we can use an oscillator, amplifier, and an electro-mechanical oscillator.  We have a great advantage, we can easily vary the frequency of excitation thereby providing a whole vista of experimentation.

A 17 in. x 14 in. guitar shape at 200 Hz

The same piece at 235 Hz.  There are many more resonances at higher frequencies

What the Plates do:

Vibrate (in multiple modes) as functions of:

• plane dimensions
• mass per area of the planes
• excitation frequencies
• locations of excitation

Why do This?:

To study the resonance conditions of the (usually) wooden parts of stringed instruments; e.g. violins, oboes and guitars – although similar studies are applied to pianos, drums, cymbals, and bells.

In Practice:

• The plate under study is (often) vibrated (and supported) at its center of gravity
• Salt is sprinkled on the plate’s surface
• Starting with the vibration at a low frequency (e.g. 100 Hz), slowly increase the frequency until a first resonance is obtained – adjust the amplitude of vibration as necessary to achieve salt migration – you should be able to hear the sound – too much amplitude will cause excessive motion of the salt (and poor patterns)
• Successive resonances are observed with the salt moving from pattern to pattern
• ‘Rock’ the frequency very slowly around a resonance point to achieve exactly the resonance frequency (sharp salt lines)

12” Square at 258 Hz.

• Increasing the driving frequency causes the salt to move into the next higher resonance patterns

At 495 Hz

At 870 Hz

At 1259 Hz

• All plate shapes will exhibit multiple resonance conditions
• Some salt will vibrate off the plate.  Use a large shaker to add salt as necessary.

Why the Patterns?:

When the plates achieve a resonance condition, ‘standing waves’ are created.  This is, in fact, analogous to the similar effect in a vibrating string – except this is in two dimensions.

At resonance, the plate’s anti-nodes will be oscillating up-and-down energizing the salt – the salt will (naturally) move towards a lower energy level.  The lower level is a node.  That’s where the salt will collect (and remain), creating the lines we see.  These are the lower energy (non-vibrating) zones.

The ‘Exciter’:

Any commercially available electro-mechanical unit will work well for this demonstration.  However, these units are expensive (>\$200).  An alternative is to build-your-own as illustrated in the book Physics Demonstration Apparatus .  Its cost is a (discarded) mid-range audio speaker, a wooden box and a construction, coupling the speaker’s cone to a vertical rod.  Building the unit, as shown in the book, does require some machined metal parts and a little ingenuity can simplify the unit’s construction (wooden pieces in place of aluminum pieces).  However, be mindful that the air’s varying humidity will affect the ‘fits’ of wooden components – that’s why the book’s design utilizes aluminum for the top plate and the rod guide.

The Plates:

Although wood and cardboard will work, both are susceptible to warping.  For this reason, I make plates of very thin steel and aluminum – typically about 1/64” (0.0156”) thick.  My sources are the (discarded) side panels of tower computers and the covers of (discarded) microwave ovens.  Don’t attempt to use sheet metal shears to cut plates from these.  Instead, use a very fine (at least 24 teeth-per-inch) band saw blade or a similarly fine-tooth saber saw blade.  These methods yield a flat surface at the periphery of the plate.  The demonstrations do require a very flat surface to produce acceptable resonance patterns.  Be sure to file the edges free of burrs.

Plates in the range of 12 inches (square/round) work quite well.

Drill a hole at the plate’s center-of-gravity.  Use a banana plug to connect the plate to the vibrating rod.

Two resonance patterns of a 12 in. diameter round disk:

At 175 Hz

At 240 Hz

Two resonance patterns of a 12 in. square with rounded corners:

At 180 Hz

At 290 Hz

In the Classroom:

This is a wonderful real-time demonstration.  And, even better, the plates can be photographed at their resonance frequencies, to be compiled into labs, reports, science projects, as either hard-copy or as PowerPoint presentations.

Endless possible plate shapes provide a great variety of investigations – different sizes of square, rectangular, round, and musical instrument shapes – enough explorations to keep several groups of students truly engaged in fascinating exercises.

Marty Sagendorf is the author of the book Physics Demonstration Apparatus. This amazing book is available through Educational Innovations and includes ideas and construction details, including all equipment necessary, for the creation and use of a wide spectrum of awe inspiring physics demonstrations and laboratory equipment.  Included are 48 detailed sections describing hands-on apparatus illustrating mechanical, electrical, acoustical, thermal, optical, gravitational, and magnetic topics.  This book also includes sections on tips and hints, materials sources, and reproducible labels.