The frequency of a pendulum is inversely proportional to the square root of its length, while the frequency of a pipe is inversely proportional to its length.
The frequency of a pendulum, f, depends on its length, L, and on the acceleration of gravity,g.
f = (1/2p) (g/L)0.5
If the length of the pendulum is cut to 1/4 of its original length the frequency of its swing will double.
The frequency of a tube open at both ends depends on the speed of sound, v.
f = v/2L
If the length of the tube is cut to 1/4 of its original length the frequency quadruples.
If you have two pendulums with lengths PL1 and PL2 and frequencies Pf1 and Pf2 with a ratio Pf2/Pf1
and you want to make two tubes with lengths TL1 and TL2 and the same frequency ratio Tf2/Tf1
Then Pf2/Pf1 = (PL1/PL2)^0.5 = Tf2/Tf1 = TL1/TL2
and TL1/TL2 = (PL1/PL2)^0.5
The ratio of the lengths of the tubes is the square root of the ratio of the lengths of the pendulums.
Scientific Explorations with Paul Doherty
21 October 2006