BIO Tendons are, essentially, elastic cords stretched between two fixed ends. As such, they can support standing waves. A woman has a 20-cm-long Achilles tendon—connecting the heel to a muscle in the calf—with a cross-section area of 90 mm^2 . The density of tendon tissue is 1100 kg/m^3 . For a reasonable tension of 500 N, what will be the fundamental frequency of her Achilles tendon?

Video transcript

Hey, everyone. So this problem is dealing with frequency in fixed strings. Let's see what it's asking us consider a guitar string made from an elastic material that can support standing waves when stretched between two fixed points. This guitar string is 14 centimeters long and has a cross sectional area of 78 millimeters squared. The material's density is 980 kg per meter. Cued when the string is under attention of 400 newtons find the lowest frequency of the guitar string. Our multiple choice answers here are a 266 Hertz B 258 Hertz C Hertz or D 245 Hertz. OK. So the first thing we can do here is recall a frequency um for a uh string with two fixed points is given by the equation N V divided by two L where N is the number of harmonics V is the speed and L is the length because we're looking for the lowest frequency, that's when N equals one because we can recall that N is always an integer. So the lowest frequency will be when N equals one. And so then we have the length uh that was given to us in the problem. And so now we need to find speed. So next, we can recall that speed in terms of tension, which was also given to us is given by the equation B equals the square root of T divided by mu where you in turn, where mu is the linear density. So in turn U equals your mass divided by your length. And while we weren't given the linear density of the string, we were given the, the material density. And so those two are, are related and we can use that material density to substitute in for a meal. So by definition, density is equal to mass divided by volumes. That's our big V not to be confused with our small V for velocity, and our volume is equal to our area multiplied by our length. And so we can um solve this equation for our mass. So we'll have row, our density multiplied by our area multiplied by our length is equal to our mass plug that into mu, the mu equation. And that gives us row A L divided by L those length to actually cancel there. And then mu is simply row A. So we plug that back in for velocity, we have square root of T divided by row A. And we do have all of those terms from the problem. So we are going to solve for speed and that will solve our frequency. So speed is equal to the square root of tension, 400 newtons divided by our density that's kg per meter cubed multiplied by our area. And so our area was given to us in millimeters squared and we need to get it into standard units of meters squared. And so that will be 78 times 10 to the negative six m because millimeters to meters is minus three or to the negative three exponents and then you square it or multiply it by two uh for exponents. And that's how you get to the negative six. So 78 times 10 of the negative six m squared is our area in standard units. And we plug that into our calculator and we get point 34 m per second. Now, when we plug that into our frequency equation, we'll have 72.34 m per second divided by two, multiplied by our length. And again, that length was given to us as 14 centimeters. So we're gonna rewrite that as 140.14 m. So a little bit of an easier conversion this time and we'll plug that in and get to Hertz. And that is the answer to this problem. When we look at our multiple choice answers, it aligns with answer choice B. That's all we have for this one. See you in the next video