Sound Waves Legacy Problem #13 Guided Solution

One of the common student difficulties with solving problems in physics is the tendency to simply ignore units, or not to understand units whatsoever. Here in this question, what we're asked to calculate is the tension of a wire. What we're given is the speed of vibrations in the wire, 253 meters per second, the length of the wire, 2.45 meters, and the mass of the wire, 19.5 grams. The equation which relates these quantities to tension is the equation that you'll find on the overview page for this set of problems. It goes the V, that's the speed, is equal to the square root of the tension divided by the mass per length. Now the mass per length is the linear density, or the mass density of the wire. It's found by taking the mass and dividing by the length, a pretty easy idea. Now in this question, we need to solve for the tension, which is inside the radical on the right side of the equation. Now solving for tension means that you need to get this equation rearranged so that it's in the form of tension equal set of variables. So to do that, I'm going to square both sides of the equation. It becomes V squared equal the tension divided by the mass per length. And then I'm going to multiply both sides of the equation by the mass per length. And the equation ends up being tension equal the mass per length multiplied by V squared. Now the V squared is pretty easy. It's 253 meters per second. I need to square it. The mass per length is a little bit more difficult, especially if you're not paying attention to units. The mass per length must be in units of kilograms per meter. Those are standard metric units. And when you do that, you'll end up getting the tension in units of newtons. So I need to take the 19.5 grams, and before I divide it by the length of 2.45 meters, I need to get it to units of kilograms. The demand is that you move the decimal place three places to the left, and you get .0195 kilograms. You take the .0195 kilograms and you divide it by 2.45 meters. The result is multiplied by 253 squared. And when you're done, you end up with 509.4594 newtons. And I'll round that to three significant digits such that the answer is 509 newtons.