So this question gives us the vibrational frequency off a mode which I've stubbed. F n equals 780 arts and then it gives us the vibrational frequency off the mode immediately following that. So I've said F n plus one is equal to you 1040 arts. It gives us the length off the string on. It tells us that the string is fixed at both ends. So part of this question asks us for the fundamental vibrational frequency given this information. So you know that the frequency of the string can be defined by the velocity off the way of divided by the So we know that the frequency of the wave could be defined by the velocity divided by the Waveland Lamba. And we also know that because the string is fixed at both ends, that Lambda is given by two times the length of the string because, as we can see, if it's fixed at both ends, we have minimum amplitude on. It will resemble something like this at the fundamental frequency. So the first mode off vibration, so therefore we can construct some simultaneous equations. We can say that the Waveland off the 10th mode is equal to to l divided by n So this is like a general case. And as such, we can also say that Lunda of n plus one is equal Teoh to l over and plus one in these two together in terms of their solutions for end. We concede it n is equal Teoh Well, we have to l over Lambda and however we can say that this lambda is equal to from this equation, we can say that our lumber is equal to the over f just by rearranging that so that's actually equal to two l f over the So we can write this out in terms of the solutions for both Lambda and Land A and plus one. So this becomes and is equal Teoh, We have two times R l which is 1.6 meters times by our frequency, which is well for the for the N case that's going to be 780 hertz on divided by the on camera suiting that ve is the same both the end on end plus one cases on so equating that to the end plus one case you want to solve this equation for end now on. In order to do that, well, we have the same times before, So that would be to L, which is two times 1.6 meters times by f of N plus one, which is 1000 and 40 hearts on. That's also over V. However, we also need to minus one because this fast term here on the right hand side, is just the solution for n plus one. So we need to minus one in order to equate this to the previous end. So we have this simultaneous equation here on do. What we can do is we can solve this. We can solve it for the and so doing that we can see that we're multiplying by V rear and simply just rearranging for it gives us that V is equal. Teoh two times, 1.6 meters times. And then it will be be difference because what we will do is we will take this of minus one to the left hand side by adding one. And then we will minus this term on the left to the inside. So that gives us multiplied by 1040 minus 780 hertz Andi that comes out as 832 meters per second. So what we can then do is go back to our first equation that our frequency is equal to the velocity over the wavelength at the fundamental frequency. So we know that that is larvae. We just worked out as 832 meters per second on our Lunder, which we went house to L, which is two times 1.6 meters. Andi, that comes out as 260 hertz, which is the answer in the textbook. So part B of this question asks us while it says that our tension in the string is equal to 1200 Newtons and asks us to work out the mass off the string. So we know that the fundamental frequency F one is defined as well. We know that it's equal to 260 hearts for one because you about in the previous question and it's going to be equal to you from the equation in the textbook one over to L multiplied by the square root of tea over new where mu is our mass per unit length so is equal to em over El Onda. We therefore want to solve for em. So rewriting that out we concede it 260 hertz which is our fundamental frequency is equal to one over to l roots Attention tee times by the length l the m because, as you can see, having em over L on the denominator will bring the L to the top of the fraction. And so rearranging for em conceive it. M is going to be equal to one over to l F one, which is all 260 hunts that will be squared because M is in the square root multiplied by our attention multiplied by our laugh. So rewriting that out in terms of the numbers that we already know that's equal to you one over two times 1.6 times 260 squared. Why units are in s I form. So you need to you worry about that multiplied by attention which is 1200 Newtons multiplied by the length of our string, which is a 1.6 meters Andi that comes out as in sa units as north points and then we have north north to eight kilograms, which is in turn equal to 2.8 grams eight grams. That is our final answer.