Hey, guys. Let's take a look at this. We're not given any numbers in this problem. It's all going to be conceptual, right? So we're saying that we're increasing the amplitude of an oscillation, and we're asked which of these statements are correct. So let's just take a look at the first one. The period of oscillation increases. Okay. So let me just go into my equations and figure out what do I have for equations of period. Well, that's going to be the big giant omega equation right here. So in this equation for omega and t and all that stuff, is there anything that involves amplitude? No, there's not, right? It's just the frequency, the period, and then k doesn't change with a and the k is just a property and then mass is just mass. So that means the period of oscillation does not increase. So that's wrong. So let's take a look at the second one. The maximum acceleration increases. Let's look at our formulas for a max. We've got 2 of them. This one, a max, is when kma. So what happens is that in this equation for a max, if a increases and k and m are just properties of the spring and the mass of the object, then that means that a max also has to increase. So that means that this is actually a correct statement. I think I've got a repeat between c and d, but whatever. Okay. So that is actually true. So I'll write that there.
Part c is asking for the maximum speed. So what happens to the maximum speed if we increase the amplitude? So just like we did over here, the v max is what they're looking for. So this v max is equal to aω. Now let's see what happens. If I increase this amplitude, does anything happen to omega? Well, omega is equal to just, you know, all of this stuff over here, and we said that that doesn't change with amplitude. So if a goes up, what happens is v max goes up. Sometimes you have to check if one of these variables will like decrease if you increase the amplitude because sometimes there might be that kind of relationship. So that's just like an extra question we have to ask. Okay. So that means that that is actually true, right? So the v max does actually increase. So that is good. So what about this, d, which I guess we'll call the maximum kinetic energy? So k max. So let's look at our energy conservation equation. Well, so energy conservation is like the maximum, elastic potential energy, whereas this is the maximum kinetic energy. So k max is when 12vmax2. So we just said that v max squared increases if you increase the amplitude, so that means that k max also has to increase. So that means, yes, it does. So sometimes you also might have those-like indirect relationships as well. So that means that does increase.
So what about the max potential energy? Okay. What does that mean? So again, we're going to look at that same equation. We said that the maximum potential energy, the elastic potential, was 12ka2. So for E, we've got that U max is equal to 12ka2. So it's pretty obvious that if this a just goes up, that means the maximum potential energy also has to increase. So that means that that is true.
And now for this last one, the maximum total energy. So maximum total energy is just going to be this whole entire mechanical energy formula. So what happens is, again, if you increase the maximum amplitude, right, if a goes up, then the whole entire mechanical energy goes up. So if mechanical energy is 12ka2 and a goes up, that means mechanical energy also goes up. So that means that that is also a true statement.
