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Acceleration of Mass-Spring Systems

Patrick Ford
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if you attach a mass to a spring it becomes a mass spring system. So now you're not pushing up against the spring itself, you're gonna take some block and push it up against the spring with some applied force. Now we know that that spring is going to push back against you in the opposite direction with some F. S. And 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 at some distance here. And so now if we 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 going to assume that the mass of the spring is equal to zero. So now we write 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 M. A. 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 or 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 is this force that's gonna want to push it or pull it back to the equilibrium. So now if we consider all these forces here we've got Fs equals M. A. Now we know that's K. X. So we have negative K. X. Whoops, 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 can just go ahead and divide over the mass and we get acceleration is equal to negative K. Over M times X. Where again this negative sign just reminds you that it's in the opposite direction of whatever 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 cake constant is equal to 15. Let me move that somewhere else. And we're told that this thing is stretched .2 m To the right beyond its equilibrium point. So we've got this deformation X. is equal to 0.2 m. So now given those two things, we're 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 write the whole equation out. We've got negative K. X. And that's equal to N times A. So if I wanted to figure out the 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 going to be negative 15, that's my spring constant and then 0.2 for the deformation. So I've got the spring force is equal to negative three newtons. Now, the reason we got a negative sign is because we're taking the right direction to be positive. So this negative sign just means it points to the left. That makes sense because once you release it, that force is going to 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 got an acceleration formula A is equal to negative K over M times X. We've got all of those numbers. So A 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, that means 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.