A Simple Physics Thought Experiment: Tuned Wood Blocks
d.w.rowlands [at] gmail.com
The following is a physics problem that came up as an example in the teaching class I took from the MIT Teaching and Learning Lab in spring 2014. I thought it was interestingly subtle and worth sharing.
Suppose you have two long blocks of wood of the same length. The only difference between them is that one is twice the thickness of the other. If you hang them from the top and tap each one on the side to ring it, which produces the higher tone, and why?
This problem doesn't depend on any particularly special properties of wood; the answer and explanation would also hold, for example, for metal blocks. The thicker block will produce the higher tone. This is somewhat counter-intuitive, since larger objects often produce lower tones because they can accommodate larger wavelengths. However, in this situation, the audible sound is produced by air responding to a lengthwise standing wave introduced in the block. Since both blocks are the same length, they can support standing waves of the same wavelength. However, the thicker block is more rigid, so it produces a stronger restoring force and a standing wave of the same wavelength in it will have a higher frequency. Thus, while both blocks vibrate with the same wavelength, the thicker block has a higher frequency and induces higher-frequency sound waves.