Consider a string with a length of 56.5 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 speed of the wave in the string, we can use the formula:
v = f * λ
where:
- v is the velocity of the wave,
- f is the frequency of the wave, and
- λ is the wavelength of the wave.
In this case, we are given the frequency f = 272 Hz.
To find the wavelength λ, we need to consider the length of the string. The first harmonic (also known as the fundamental frequency) occurs when the wavelength is twice the length of the string. So, we can calculate the wavelength using the formula:
λ = 2L
where L is the length of the string.
In this case, the length of the string is given as 56.5 cm. However, we need to convert it to meters to match the units for the velocity.
1 cm = 0.01 m
So, L = 56.5 cm * 0.01 m/cm = 0.565 m
Now we can substitute the values into the formula for velocity:
v = f * λ
v = 272 Hz * 2 * 0.565 m
Calculating this, we get:
v ≈ 306.096 m/s
Therefore, the speed of the wave in the string is approximately 306.096 m/s (rounded to 3 significant figures).