. A flute is designed so that it plays a frequency of 261.6 Hz, middle C, when all the holes are covered and the temperature is 20.0o C. (a) Consider the flute to be a pipe, open at both ends, and find its length, assuming that the middle-C frequency is the fundamental. (b) A second player, nearby in a colder room, also attempts to play middle C on an identical flute. A beat frequency of 3.00 Hz is heard. What is the temperature of the room

Explanation:

The first case relates to open end pipe with fundamental frequency . Let the length of flute be l . The length vibrating column  is l .

λ = v / n where n is frequency , v is velocity of light and λ is wavelength of sound produced .

λ = 343 / 261.6

= 1.31 m .

For fundamental frequency

λ / 2 = l

l = λ / 2

= 1.31 / 2

= .655 m

= 65.5 cm .

b )

Nearby colder room will create lower frequency because velocity of sound will be smaller .

frequency of note produced = 261.6 - 3 = 258.6 Hz .

velocity of sound v = n λ

= 258.6 x  1.31 m

= 338.76 m /s

decrease of velocity = 343 - 338.76 = 4.24 m /s

1 degree change in temperature produces a change of .61 m/s change in velocity .

decrease in temperature = 4.24 / .61

= 7⁰C

Temperature of colder room = 20 - 7 = 13⁰C .