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 going to take some block and push it up against the spring with some applied force. Now we know that the spring is going to push back against you in the opposite direction with some FS. And we know that those two forces are equal to each other, except for this minus sign here. So we say that FS is equal to negative FA, and that's equal to negative k times x. So all that means is you're pushing up against something and you're compressing it some distance here.
And so now if we consider all the forces that are acting on this object, we can use f=ma to figure out what's going on. 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 0. So now we've read all the forces, we've got negative FA, and then we've got the positive spring force, and those two things are equal and opposite, so they're going to cancel out. That means ma is equal to 0. 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? 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 kx. So now the spring force is the force that's going to want to push or pull it back to the equilibrium. So now if we consider all these forces here, we've got FS=ma. We know that's kx, so we have −kx=ma. This is a powerful formula.
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 reminds you that it's in the opposite direction of wherever you're pushing or pulling it.
So, let's check out an example. In this example, we've got a 0.60 kilogram block that's attached to some spring. The k constant is equal to 15. And we're told that this thing is stretched 0.2 meters to the right beyond its equilibrium point. So we've got this deformation x is equal to 0.2 meters. So now, given those two components, we're supposed to figure out the force that's acting on this object and also its acceleration.
For the force on the block, that's going to be the spring force. So let's write the whole equation now. We've got −kx and that's equal to ma. So if I want to figure out the spring force here, all I need is the compression distance or the stretching distance and the force constant, and I have both of those. So I've got the spring force is equal to negative 3 N. The reason we got a negative sign is that 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. Let's just go ahead and use our formula: A=−k/mx. We've got all of those numbers. So a is just equal to −15/0.6⋅0.2, and we get an acceleration equal to −5m/s². Again, this negative sign just means that it points to the left. That makes sense because 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, 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 going!