The two highest-pitch strings on a violin are tuned to 440 Hz (the A string) and 659 Hz (the E string). What is the ratio of the mass of the A string to that of the E string? Violin strings are all the same length and under essentially the same tension.

Video transcript

Welcome back, everyone. We are making observations about a guitarist who is tuning his acoustic guitar. Now, we are told that the frequency of the fifth string or the A string is to be tuned to 110 Hertz. And we are told that the frequency of the second string or the B string is to be tuned to 246 Hertz. Now considering that the strings have the same length subject to nearly equal tension. We need to estimate the mass ratio of the A string to the B string. So how are we gonna go about doing this? Well, we know that the frequency or a standing wave is equal to the mode number times the speed of sound through the current medium divided by two times the cavity length. But we can expand this. We can actually say that the speed of sound is going to be given by the square root of tension divided by the linear density of our string. And the linear density of our string is simply just the mass divided by the length. So let's sub in the value from mu into our equation for V and then sub in V into our equation for a standing wave. What we get is that the frequency for a standing wave is, is our mode number divided by two times the length divided by our tension times length all divided by our mass. So what does this mean for our strings? Well, what we can do is we can define or rewrite this equation for each of our strings. So we have the frequency of A is M over two L divided by T S L over the mass of A. And the frequency of B accordingly is gonna be the mode number divided by two L times the square root of T sub S L divided by the mass of B. And I'm actually gonna divide the frequency of B equation by the frequency of A equation. And what we get is that everything will cancel out except the square root of our desired mass ratio. In order to isolate our desired mass ratio, we are going to square both sides of our equation. And what we get is that our desired mass ratio is equal to the frequency of B divided by the frequency of A squared. Let's go ahead and plug in those values. We have 2 46 divided by 1 10 squared which when you plug into our calculator gives us a final answer of five corresponding to answer choice. A thank you all so much for watching. I hope this video helped. We will see you all in the next one.