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Anderson Video - Period of an Oscillating Spring

Professor Anderson
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>> Hello class. Professor Anderson here. And let's talk a little bit more about simple harmonic motion. And let's ask the following question. Let's say that we have a block on a spring. And we're going to stretch it out from x equals 0 to some x equals A. And then we're going to let it go. We know what happens. It goes to there. It compresses the spring. It's going to come back out to x equals A. And it's just going to back and forth. This is the wall. Let's now do the following. Let's take that spring and instead of stretching it out to x equals A, let's just stretch it out to A over 2. And now we'll let it go from here. And now it's only going to compress the spring to negative A over 2. And it's going to go back and forth. And there is some characteristic period, T, for this behavior. And there is some characteristic T for this behavior. And let's see if we can understand how those two relate. This one has period T1. This one has period T2. And let me ask you guys. How does T1 relate to T2? Anybody have a thought? Dan. Somebody hand the microphone to Dan. Dan's properly dressed in his San Diego State attire. So we want to get that on camera as much as possible. Dan, what do you think? How do the periods relate in this case versus this case? Tell me your thoughts. Anything you're thinking. >> (student speaking) Would T2 be half as much? >> Okay. Why would you say that T2 is half as much? >> (student speaking) Because A is divided by 2. >> Yeah. Because A is divided by 2, right? It seems like it doesn't have to go as far, right? This one had to go from A all the way to negative A. And this one had to go from A over 2 to negative A over 2. So it clearly didn't have to go nearly as far, right? If it doesn't have to go as far, it seems like the period should probably be shorter, I guess. Okay? But let's see if we can relate the period back to something that we know about this motion, right? If this is a mass, m. And this is a spring, k, we just wrote down what the period is for that motion. The period T was 2 pi square root m over k. So, what do you think, Dan? Is T2 still going to be half of T1? Or is T2 going to be twice T1? Or 4 times? Or something else? What do you think? >> (student speaking) They'd be the same. >> They'd be exactly the same. Right? There is no A in there. There's no amplitude dependence. And if there's no amplitude dependence in there, then the period is in fact exactly the same. So T1 is in fact equal to T2. Now, how does that make sense with what we just talked about? This one clearly is not moving as far as this one. But what is this one doing that this one is not doing? Is this one moving faster than this one, Dan? What do you think? >> (student speaking) Yeah. >> Yeah. Of course. If I stretched the spring out really far, we know that there is a very big restoring force on it. This one's going to move a lot faster than this one. And it moves just fast enough to cover that distance in the same amount of time as this one to cover that distance. And that is sort of built in to the Hook's Law, f equals negative kx. All of it comes out that the period is independent of the amplitude. Which is kind of cool, I think.