A 1.6-m-long string fixed at both ends vibrates at resonant frequencies of 792 Hz and 990 Hz, with no other resonant frequency between these values.
(a) What is the fundamental frequency of this string?
(b) When the tension in the string is 1165 N, what is the total mass of the string?
Honestly, I am confused about this. COuld someone explain it to me. Thank you.
Sure! Let's start by understanding the concept of resonant frequencies. Resonant frequencies are the natural frequencies at which an object vibrates with maximum amplitude when excited. In the case of a vibrating string, the resonant frequencies are determined by its length, tension, and mass per unit length.
(a) To find the fundamental frequency, we need to determine the resonant frequency with the lowest value. In this case, we have two resonant frequencies: 792 Hz and 990 Hz. Since there are no other resonant frequencies between these values, the lower frequency of 792 Hz must be the fundamental frequency.
(b) To calculate the total mass of the string, we can use the formula for the speed of waves on a string:
v = √(T/μ)
where v is the speed of waves, T is the tension in the string, and μ is the mass per unit length of the string.
We can rearrange the formula to solve for μ:
μ = T/(v^2)
Given the tension T = 1165 N and the fundamental frequency v = 792 Hz (which corresponds to the first resonant frequency), we can substitute these values into the formula to find the mass per unit length, μ:
μ = (1165 N) / (792 Hz)^2
Now, since we know the length of the string is 1.6 m, we can calculate the total mass of the string by multiplying the mass per unit length by the length:
Total mass = μ * length
Substitute the value of μ into the equation to find the total mass of the string.