Welcome back everyone. So in this problem, we have a wave function that's given to us, we have Y of X and T equals six millimeters. And then we have some numbers here, right? We have five and then the the units for this radiance per millimeters and then 600 radiance per second. Some of the general sort of form of a wave function is that the first number is the amplitude and the units this is gonna be the K value, right? The thing goes in front of our X and this is gonna be the omega value. All right. So what we want to do is we want to figure out how long it takes for something to happen, how long it takes for a given particle on the string to travel between plus six and negative six millimeters. So I sort of want to like visualize what's going on here. If you take a string and sort of whip it up and down like this, you're gonna get some sort of a sine wave and the particles in the string remember are moving up and down. Now, in this case, the amplitude of this wave is six millimeters. So basically what is, what's gonna happen is that the particles on this uh on the string are gonna sort of bobble up and down between positive six and negative six millimeters over time. We're asked to find how long it takes for it to do that, right? So a given particle in the string that's gonna sort of bounce up and down between these two points that crest in the trough forever. That's always when it's gonna happen. And we're trying to figure out how long it takes for that to happen. So delta T, all right. So how do we solve for this? Well, the basic relationship between delta T and velocity and displacement is that V is equal to delta X over delta T, right? So displacement over time. So if I solve with this equation here, I can rearrange this and delta T equals delta X, the displacement divided by velocity. All right. So just very simply here, if I want to figure out time, I need to figure out the displacements, what's the distance of these particles are traveling divided by, what's the uh wave speed or what's the, the sort of trans velocity um of the particles that are bobbing up and down? All right. So how do I figure this out? What's the displacement? Well, the displacement really is just going to be the distance between the top of the crest and the bottom of the trough. And it's really just the distance between plus six and minus six. In other words, it's basically just double the amplitude. So this is really just gonna be two times amplitude, which is just gonna be 12 millimeters. All right. Now, I can also convert this to 0.012. But that's basically where the displacement is. All right. So that's done. So all I really need now is I need to figure out now the velocity, the display the velocity of the particles that are moving up and down on the string. And we have a new equation for this. It's the transverse velocity or whatever. Um And basically the equation for the velocity is it's going to be the uh oh I'm sorry, it's gonna be the um the displacement velocity of the particles which is going to be omega divided by K. All right. So if we do do Omega divided by K, uh basically, I'm just gonna take this uh omega which is 600 the units for this are gonna be really important. So notice I have 600. This is gonna be radiance per second divided by and this is gonna be five and this is gonna be radiance per millimeter here. All right. So this is important because what happens is the radiant are gonna cancel when you do this. And what you end up getting is you end up getting 100 and 20 millimeters per second. So the units are gonna be really important here. So this is what my velocity is. It's 100 and 20 millimeters per second. All right. So then how do I figure it? Now? Delta T? Well, really, now I have everything I need to solve because now I have delta X and I have V. So I'm just gonna bring this down here and my delta T is just gonna be my delta X, the displacement which is 12 millimeters divided by, and this is gonna be 100 and 20 millimeters per second. So again, the units are important here because if you got something like meters and millimeters, you're gonna have to convert. But what we're gonna see here is that millimeters will cancel and you're just gonna be left with seconds. And really what happens is this is actually just gonna be 0.1 seconds. So in other words, it takes 1/10 of a second or 0.1 seconds or a particle in the string to sort of bounce up and down between positive six and negative six. That's will always happen. All right. So it's kind of a strange problem, but we're really sort of pulling together a lot of different equations from, from wave functions. Let me know if you have any questions. Thanks for watching.