## 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.