Consider a string with a length of 56 cm tied at both end (like on a stringed instrument). If the frequency of the first harmonic on the string is 272 Hz, determine the speed of the wave in the string. Post your answer in m/sand with 3 significant figures.
To determine the wave speed in the string, we can use the formula:
v = fλ,
where v is the wave speed, f is the frequency of the wave, and λ is the wavelength.
In this case, we are given the frequency of the first harmonic (f = 272 Hz) and we need to find the wavelength. For a string with both ends tied, the wavelength of the first harmonic is twice the length of the string, so:
λ = 2L,
where L is the length of the string.
Given that the length of the string is 56 cm, we first need to convert it to meters:
L = 56 cm = 0.56 m.
Substituting the values into the equation, we have:
λ = 2 * 0.56 m = 1.12 m.
Now we can plug the values for frequency (f = 272 Hz) and wavelength (λ = 1.12 m) into the wave speed formula:
v = fλ = 272 Hz * 1.12 m = 304.64 m/s.
Therefore, the speed of the wave in the string is approximately 304.64 m/s (rounded to 3 significant figures).