To calculate the lengths of the pendula, start with the longest time
period, T. Decide how many cycles of that pendulum you would want
before all the pendula are back in phase, usually 12 to 20 cycles.
Let us use 16 cycles. This means that the time period (Ta) of the
next pendulum needs to be such that it takes 17 cycles to get back
in phase with the first, the next will take 18 cycles, the next 19,
etc...
This
leads to the relationships: 16T = 17 Ta, 16T = 18 Tb, and 16T =
19 Tc, etc.
To
calculate the lengths, form the ratio of time periods as a function
of lengths, then solve for the ratio of lengths as a function of
time periods. When you are finished this exercise for the student,
the result shows that the length you are looking for is the length
of the longest pendulum multiplied by the square of the ratio of the
time periods squared.
For example: La = L (16/17)^{2},
Lb = L (16/18)^{ 2}, Lc = L (16/19)^{ 2}, etc....
where L is the length of the length of the longest pendulum.
When you decide how long you want the
longest pendulum to be, the lengths of the others can be calculated
easily. (Spreadsheets are great for this!). To build the stand,
calculate the difference in lengths for adjacent pendula. Take the
difference between adjacent pendula, this will form the steps on the
stand so the swinging bobs are at the same level. The method of
attachment is up to the builder, but the string supporting each
pendulum needs to be pinched to make a definite length. The bob
support string is bifilar, or two stringed, one on each side of the
bob. The top ends need to be separated by enough distance to assure
stability. The ability to make fine adjustments to the length is
necessary so the device can come back to line after two or even
three cycles.
One design for a 20 cm longest
pendulum is to cut 5.0 cm long blocks of 1 by 2 pine and sand the
sides so they will glue together firmly. I then screw on a 1 cm
wide strip of hard board across the bottom so that the edge of the
strip lines up with the bottom of the block. The strings to the
bobs goes under the strip and is held in place by the screw. Using
the difference in length between the adjacent pendulums as the
offset, glue the blocks together.
For the PTRA workshop at the 2009
Summer Meeting of the AAPT at the University of Michigan, I scaled
down the Pendulum Wave to the over-head projector size. When the
pivot board is in the center of the stage, it does not affect the
image on the screen, and the motion of the bobs is clearly seen.
See the video on YouTube at:
(http://www.youtube.com/watch?v=WDHUP3J-ebc) |