Acceleration of Mass-Spring Systems

by Patrick Ford
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Hey, guys. So now that we've seen Hook's law in the equation for Forces of Springs, we want to see what happens if you attach an object with some mass to a spring. Let's check it out. So if you attach a mass to a spring, it becomes a mass spring system. So now you're not pushing up against this spring itself, you're gonna take some block and push it up against this spring with some applied force. Now we know that that spring is gonna push back against you in the opposite direction with some F s. We know that those two forces are equal to each other except for this minus sign here. So you say that F S is equal to negative f A. And that's equal to negative K times X. So that means that you're pushing up against something and you're compressing it some distance here. And so now if you consider all the forces that are acting on this object, we can use f equals m A to figure out what's going on so that the first thing is that this M always refers to the mass of the object itself. And so we're always gonna assume that the mass of this spring is equal to zero. So now if you read all the forces, we've got negative f a and then we've got the positive spring force and those two things are equal and opposite, so they're going to cancel out. So that means Emma is equal to zero. And so therefore, if you're just pushing up against this thing and keeping it there, there's no acceleration. So now what happens if I release that applied force? So if I remove my hand now, the only thing that's pushing up against the spring air this object here is that spring force, the Applied Force goes away, and so that's equal to negative K X. So now the spring Force is the sport. Is this force that's gonna wanna push it or pull it back to the equilibrium? So now, if we consider all these forces here, we've got f s equals M A. Now we know that's K X. So we have negative K X negative. K X is equal to mass times acceleration. This is a really, really, really powerful formula. And so now if we want to solve and calculate for the acceleration. We could just go ahead and divide over the mass and we get acceleration is equal to negative K over em. Times X where again this negative sign just reminds you that it's in the opposite direction of wherever you're pushing or pulling it. So let's check out an example. So this example here, we've got a 0.60 kg block that's attached to some spring here. We've got the cave. Constant is equal to 15. Let me move that somewhere else. And we're told that this thing is stretched 0.2 m to the right, beyond its equilibrium points. So we've got this Deformation X is equal to 0.2 m. So now, given those two things were supposed to figure out the force that's acting on this object and also its acceleration. So in this, we're being asked for the force on the block. So that's gonna be the Spring force. So let's right the whole equation. Now we've got negative K X, and that's equal to M times A. So if I wanted to figure out this spring force here, all I need is the compression distance or the the stretching distance, and the Force Constant and I have both of those, so it's gonna be negative. 15. That's my spring constant and then 0.2 for the deformation. So I've got this spring Force is equal to negative three Newton's now. The reason we got a negative sign is because we're taking the right direction to be positive. So this negative fine just means it points to the left. That makes sense, because once you release it, that force is gonna be acting in that direction. So now we're supposed to find the acceleration. So let's just go ahead and use our formula. We've got an acceleration. Formula A is equal to negative K over m times X. We've got all of those numbers so is just equal to negative 15 divided by 0.6 times 0.2 and we get an acceleration that's equal to negative five meters per second squared again. That negative sign just means it points to the left. That makes sense, because if this is the only force that's acting on this thing, I mean the acceleration must be towards the left. If you ever forget this formula here, you can always get back to this just by using the spring force is equal to mass times acceleration. Those two things are equal to each other. Alright, guys, that's it for this one. Let's keep