A 0.28-meter long spring with a force constant of 25 newtons per meter is compressed by a force of 1.25 newtons. What is the length of the compressed spring?
Okay, so in order to answer this question, let’s start out by drawing a diagram of the spring. So here’s the spring before it’s compressed. Now, we’ve been told that the length of the spring is 0.28 meters. Next, we’re told that a force of 1.25 newtons is applied to the spring and this results in the compression of the spring. So here’s the compressed spring with the 1.25-newton force applied to it.
What we need to do is to find out the length of the compressed spring. In other words, we need to find out this length here, which we’ll call 𝑙 sub 𝑐 for the length of the compressed spring. As well as this, we know the force constant of the spring. This value which we’ll call 𝑘 is 25 newtons per meter. So how we’re gonna go about working out the value of 𝑙 sub 𝑐? Well to do this, we can use something known as Hooke’s law.
Hooke’s law tells us that the force 𝐹 that is applied to the spring is equal to the force constant of the spring multiplied by the extension of the spring. Now, if the extension of the spring is positive, then the spring elongates and if the extension is negative, then the spring is being compressed.
In other words, if the original spring is extended so it moves towards the right, then the extension is positive and the force is being applied to the spring is also towards the right. However, if we apply a negative force or in other words a force towards the left, then the spring will compress resulting in a negative value for the extension. Now, the extension of the spring or in this case the compression is this distance here, which we’ll call 𝑥. And we can calculate 𝑥 using Hooke’s law.
So let’s rearrange the equation. If we divide both sides by 𝑘 the force constant, then the 𝑘s on the right-hand side cancel, leaving us with the following expression: the force applied to the spring divided by the force constant is equal to the extension. Now, as we’ve already mentioned, forces towards the right are positive and forces towards the left are negative. And the same is true for the extension — towards the right is positive and towards the left is negative.
Bearing that in mind, we can substitute in the values of the force and the force constant. The force is negative 1.25 newtons because it’s to the left and the force constant is 25 newtons per meter. Now, when we evaluate this fraction on the left-hand side, we get a result of negative 0.05 meters and that’s our value of 𝑥 — the extension. However, remember the negative sign only tells us that the extension is towards the left. The spring is being compressed. The actual magnitude of the extension 𝑥 is just 0.05 meters. In other words, the length by which the spring is compressed is 0.05 meters. And so this distance is 0.05 meters.
At this point, we can now work out the length of the compressed spring. So now, we can say that 𝑙 sub 𝑐, the length of the compressed spring, is equal to the length of the original spring minus the compression of the spring. Because the spring is being compressed, so it’s becoming shorter. So the value of the length of the spring after it’s compressed is going to be smaller. Hence, we subtract 0.05 from the length of the uncompressed spring.
Now, another way to think about this is that we’re adding an extension of negative 0.05 meters to the original length of the spring. In other words, the final length of the spring is equal to the original length plus the extension which is negative 0.05 meters. But that’s just convention and it’s one way to deal with negative signs. But by far, the best way to deal with negative signs is to draw a diagram of the situation. This way you can visualize what’s going on and you don’t need to worry about what negative goes where.
In this case, you can use your understanding that when the spring is compressed, its length is going to get shorter. But anyway, so the value of 𝑙 sub 𝑐 therefore ends up being 0.23 meters. And hence, our final answer is that the length of the compressed spring is 0.23 meters.