A 1.0-m-tall vertical tube is filled with 20°C water. A tuning fork vibrating at 580 Hz is held just over the top of the tube as the water is slowly drained from the bottom. At what water heights, measured from the bottom of the tube, will there be a standing wave in the tube above the water?

Video transcript

Hey, everyone. So this problem is dealing with sound waves. Let's see what it's asking us. A 450 Hertz pure sound wave is directed toward the upper end of a graduated glass cylinder held straight up the glass cylinder of 130 centimeters in height is totally filled with oil and has a tap at the base. The oil is at room temperature. The oil is drained gently from the tap. We are expecting that standing waves will be generated inside the cylinder at specific oil levels denoted here by H, the level H is measured from the base, calculate the values of H. So our multiple choice answers here are a 1. times 10 to the second centimeters and 7.3 times 10 to the first centimeters. B 7.3 times 10 to the first centimeters and 1. times 10 to the first centimeters. C 1.1 times 10 to the second centimeters, 7.3 times 10 to the first centimeters and 3.5 times 10 to the first centimeters. Four d 1.1 times 10 to the second centimeters, 3.5 times 10 to the first centimeters and 1.5 times 10 to the first centimeters. Ok. So the first thing I'm going to do here is actually kind of draw what's happening. So you can better understand what the problem is asking. So we have this graduated cylinder and it's completely filled with oil. We have oil at the top initially and we know that the total height of the um, cylinder is 130 centimeters, which is 1.3 m, there's a tap at the bottom of the cylinder. And so as oil is drained out, our level at the top comes down to various lots. So as the oil drains down, what's actually happening is we are creating an open closed tube system where the closed end is the oil. And so the various heights of the oil as it drains, that's what we're trying to find where we'll have those sound waves. Ok. So as far as our uh standing wave equations in an open close two, we can recall that frequency is given by N V divided by four L. And so we need to find what else create those weights. We're called again that in a um open, close tube, open, close pipe system, um our ends are odd integers. So N can only be 1357, et cetera. So let's look at uh this equation. If we rearrange it for L, we have L equals N multiplied by V all divided by four. So we'll have N B is the speed of sound in air at room temperature. We can recall that's 343 m per second divided by four. And then F was given to us the problem and the frequency of 450 Hertz. And so what we're going to do here is just solve this equation for the first few integers of N. So N equals 135 and see what that gets us. So when we plug in this equation, we, when we plug in one for N, we get 0.191 m. When we plug in three, for N, we get 0.572 m when you plug in 54 and we get 0.953 m. When we plug in seven for N, we get 1.33 m. Now that length, 1.33 m is larger than the total height of the um cylinder. So we know that that is not an option. And so we know that we have three standing waves at these lengths. And the last maybe kind of tricky part is that this uh equation is calculating L as far as the length of the open ended pipe. So that's the length from the top to wherever the level of oil is. Whereas h the problem tells us is measured from the bottom. And so what that means is that the last step here, his age is equal to 1.3 m minus L and so we can plug these L values in. So we'll have H one is 1.11 m. H sorry, H three is equal to 0.7 to eight m and H five is equal to 1347 m. And now we look at our multiple choice answers and they actually um are presented in terms of centimeters and scientific notation. So we will rewrite these. So this would be 1.1 times 10 to the second centimeters. This will be 7.3 times 10 to the one centimeters and age five will be 3.5 times 10 to the one centimeters. And that is the final answer. These three heights are the final answer to this problem. When we go back and look at our multiple choice answers that aligns with answer choice C so C is the correct answer for this one. So that's all we have for this problem. We'll see you in the next video.