Question

Identical cellos are being tested. One is producing a fundamental frequency of 248 Hz on a string that is 1.06 m long and has a mass of 100.0 g. On the second cello the same string is fingered to reduce the length that can vibrate. If the beat frequency produced by these two strings is 4.0 Hz, what is the vibrating length of the second string? (Express your answer to the nearest hundredth of a meter.)

_____m

I got 1.13m, but it said it was incorrect. Please help, thanks.

Answer 1

We have two cellos, #1 and #2. The first cello has a vibration frequency f1 = 248 Hz. To produce a beat note of 4 Hz with the first cello means the second cello must be operating at frequency f2 = 248 ± 4 Hz . We also know that fingering the cello string produces a higher frequency, not a lower one, so the second cello must be operating at f2 = 252 Hz.

The frequency of a vibrating string on a cello is given by:

f = 1/2L(T/μ)^1/2 where T is the string tension, μ is the linear mass density of the string, and L is the length of the string.

In this case, T and μ are assumed to be invariant as the cello string is fingered, so the only change is the length of the string of cello 2. The ratio of the vibrating frequencies is then:

f1/f2 = [1/2*L1 * const]/[1/2*L2 * const]
f1/f2 = L2/L1

L2 = 248/252 * L1
Since L1 = 1.06 m
L2 = 1.04317 m