2015 AP Physics 1 free response 5

The figure above shows a string with one end attached to an oscillator and the other end attached to a block. There's our block. The string passes over a massless pulley that turns with negligible friction. There's our massless pulley that turns with negligible friction. Four such strings a, b, c, and d are set up side by side as shown in the figure in the diagram below. So this is a top view. You can see the oscillator. This is the top view of the oscillator string pulley mass system, and we have four of them. Each oscillator is adjusted to vibrate the string at its fundamental frequency, f.

So let's think about what it means. I'll keep reading, but then we'll think about what fundamental frequency means. The distance between each oscillator and the pulley, l, is the same. So the length between the oscillator and the pulley is the same, and the mass of each block is the same. So the mass is what's providing the tension in the string. However, the fundamental frequency of each string is different. So let's just first of all think about what the fundamental frequency is, and then let's think about what makes them different.

The fundamental frequency, one way you could think about it, is it's the lowest frequency that is going to produce a standing wave in your string. So it's the lowest frequency that produces a standing wave that looks like this. It's a standing wave where the string is half a wavelength. There are two ways to think about it: it's the lowest frequency where you could produce a standing wave, or it's the frequency at which you're producing the standing wave with the longest wavelength.

So the string at the fundamental frequency is just going to vibrate between those two positions. You see here that the wavelength is twice the length of the string. If you wanted to see that a little bit clearer, if I were to continue with this wave, I would have to go another length of the string in order to complete one wavelength. Another way to think about it is you're going up, down, and then you're going down back. It's reflecting back off of this end here.

As we mentioned, essentially what the mass is doing is providing the tension. The force of gravity on this mass is providing the tension in this string. The oscillator is vibrating at the right frequency to produce the lowest frequency where you can produce this standing wave.

So let's answer the questions now. We have four of these setups, and they all have different fundamental frequencies. The equation for the velocity, v, of a wave on a string is v = √(T/m/l), where T is the tension of the string and m/l is the mass per unit length (linear mass density) of the string. Hopefully, this makes sense. It makes sense that if the tension increases, your velocity will increase.

You can think of the atoms of the string and how much they're pulling on each other. If there's higher tension, well, they're going to be able to move each other better. As the wave goes through the string, it also makes sense that the larger the mass (if you hold everything else equal), that you're going to have a slower velocity. Mass is, you could view it as a measure of inertia; it's how hard it is to accelerate something.

Especially if the string itself, the mass per unit length, if there's a lot of mass, it makes sense that for a given amount of the string, it's going to be harder to accelerate it back and forth as you vibrate it. This part right over here would be inversely related to the velocity. It's not proportional, though; you have this square root here. But there's definitely a relationship: if the linear mass density increases, then you're going to have a slower velocity, and if your tension increases, you're going to have a higher velocity.

Hopefully, this makes some intuitive sense. They ask, "What is different about the fourth string shown above that would result in having different fundamental frequencies? Explain how you arrive at your answer," and then part b: "A student graphs frequency as a function of the inverse of the linear mass density. Will the graph be linear? Explain how you arrived at your answer."

Alright, let's answer each of these.

So part a: What is different about the four strings because they all have different fundamental frequencies? The fundamental frequency, let's just go back to what we know about waves, that the velocity of the wave is equal to the frequency times the wavelength of the wave. Or you could say you could divide both sides by λ and say that the frequency of a wave is equal to the velocity over the wavelength.

If we're talking about the fundamental frequency, it's going to be the velocity of our waves divided by the wavelength, which is going to be twice the length of our string divided by 2l. If you look at the expression that they gave us for the velocity of the wave on the string, well this is going to be equal to √(T/(m/l)) and then all of that is going to be over 2l.

Now, all of them have different fundamental frequencies, but let's think about what's different here. They all have the same tension. How do I know that? Well, what's causing the tension is the mass hanging over the pulleys, so the weight of those masses is going to be the same for all of them. And they all have the same length; they all have the same length.

So these are all the same. The only way that you're going to have different fundamental frequencies is if you have different masses. So different masses have to be different. To answer their question, the strings must have different mass. They would have different linear mass densities, but since they're all the same length, they would also have different masses.

So let me write this down: Strings must have different masses and mass densities since all other variables driving fundamental frequency are the same.

Alright, let's tackle part b now. A student graphs frequency as a function of the inverse of linear mass density. Will the graph be linear? Explain how you arrived at your answer.

So student graphs frequency. Let me write, let me underline this. They're graphing frequency as a function of linear mass density. So we actually can write this down.

So if we wanted to write frequency as a function of linear mass density, we could write it as a function of m/l. Well, this is going to be equal to, we could rewrite this expression. Actually, we could just leave it like this. This is the same thing as (1/(2l)) √(T) / (m/l), or you could view this as being equal to √(T/(2l)).

I'm putting all of this separate because we're doing it as a function of our linear mass density. So we can assume that all of this is going to be a constant and then times √(1/(linear mass density)). So if you're plotting frequency as a function of this, it's definitely not going to be linear. You see here, I have the reciprocal of it and then I take the square root of it.

So let me write this down: this graph of f with respect to m/l is definitely not linear. I could point out that it has a reciprocal and it's a square root; it involves both the square root and the reciprocal of the variable.

So definitely not linear.

Alright, let's tackle part c. The frequency of the oscillator connected to string d is changed so that the string vibrates in its second harmonic. On the side view of string d below, mark and label the points on the string that have the greatest average vertical speed.

So one way to think about the fundamental frequency—that's our first harmonic. If they're talking about the first harmonic, and the string is what I showed before—that's the lowest frequency that produces a standing wave, or it's the frequency that produces the largest standing wave, I guess you'd say, the one with the largest wavelength.

If it was the first harmonic, the parts of the string that move the most are going to be at the center. But our second harmonic is the next highest frequency that produces a standing wave. That's going to be a situation in which the wave, instead of having a wavelength twice the length of the string, is going to have a wavelength equal to the length of the string.

Now it's going to look like this. It's going to vibrate between these points. When you see this version of it, where now we have half the wavelength, the wavelength is exactly the length of l this time. The part that moves the most is here. So that's going to move the most, and here. I didn't draw it perfectly, but one way to think about it's exactly 1/4 and 3/4 of the way.

Exactly halfway is not going to move as much; it's not going to move because it's a standing wave now. It's going to move very imperceptibly. These two places where you're going to move the most at the second harmonic—the second harmonic being the next highest frequency that produces a standing wave.