A spring with a constant of 700 newtons per meter is stretched when a force of 35 newtons is applied to it. And its length when stretched is 0.48 meters. What is the spring’s length when it is not extended?
Okay, so in this question we’ve got a spring. And we know that it’s got a constant, a spring constant, of 700 newtons per meter. We also know that it’s stretched when a force of 35 newtons is applied to it. And that its length when stretched is 0.48 meters. We need to find out the spring’s length when it is not extended. So let’s draw a diagram of the spring when it is extended, when we’ve applied the force to it.
Now here’s our spring, and here’s the spring constant of the spring. We’ve been told that this value is 700 newtons per meter, and we’re labelling it as 𝑘. We also know that the spring has been stretched. We’ve applied a force, in this case 35 newtons, and let’s say that it’s towards the right. In other words, let’s assume that this end, the left end of the spring, is firmly mounted. And we’re applying a force to the right, the 35-newton force, in order to extend the spring.
Finally, we also know the length of the spring when it’s in its stretched position. Let’s call this length 𝑠 for the stretch length. Now let’s very quickly discuss the force exerted on the spring, the 35 newtons, to the right. Well, we know that when we extend a spring, the spring itself exerts a force to resist this extension. In other words, when we exert a force to the right on the spring to try and extend it, the spring exerts a force to the left to try and resist this.
Now we’ve also been told that the length of the spring when it’s stretched is 0.48 meters. This means that the length of the spring is no longer changing. The only way that this can happen is if the forces on the spring are balanced. Because if the forces are not balanced, then the spring will not be in equilibrium. It will either continue to stretch if the force to the right is larger or it will shrink if the force to the left is larger.
However, we’ve been told that the length of the stretched spring is 0.48 meters. So the force to the left must be 35 newtons as well. This way, the forces are balanced. Now let’s call this force to the left 𝐹. This force is the force exerted by the spring itself trying to resist this extension. Now there’s a relationship that we can look at that tells us how the force exerted by the spring depends on the spring constant and the extension of the spring.
Hooke’s law tells us that the force exerted by the spring is equal to the spring constant multiplied by the extension of the spring. Now it’s the extension of the spring that can get a little bit confusing sometimes. People often trip up on what the extension actually means. So to understand this, let’s draw the spring when we haven’t applied any force to it.
Here’s the spring at its natural length. It’s not extended. It’s not stretched. And in this situation, let’s say it has a length 𝐿. We don’t know what this length 𝐿 actually is. And coincidentally, this is the length that we’re trying to find out in the question. The 𝑥 in Hooke’s law, the extension, does not refer to this length. In fact, the extension of a spring is the length of an extended or stretched spring minus the natural length of the spring.
So in the diagram, that’s this length here. How much does the spring extend by? And this is the length 𝑥. So always be very careful when defining the extension of the spring. It’s never the length of the spring, regardless of whether it had its natural length or its extended length. The extension is simply how much the spring has extended by. Anyway, so we can use Hooke’s law to calculate what this extension 𝑥 is.
And since we know the total length of the spring when it’s extended, 𝑠, we can work out the length of the spring when it’s not extended, because we know that the value of 𝑠 is equal to the natural length of the spring plus the extension. So if we know what 𝑠 is, which we do already, and if we work out what 𝑥 is, which we can do using Hooke’s law, then we can calculate what 𝐿 is. So let’s find out what 𝑥 is.
Hooke’s law tells us 𝐹 equals to 𝑘𝑥. But we want to solve for 𝑥. We can do this by dividing both sides of the equation by 𝑘. The 𝑘s on the right-hand side cancel, which leaves us with 𝐹 over 𝑘 is equal to 𝑥. We can then plug in the values of the force and the spring constant so that 𝑥 is equal to 35 newtons, the force, divided by 700 newtons per meter, the spring constant. And this gives a value of 𝑥 as 0.05 meters.
So, as we said earlier, we know what 𝑠 is and now we know what 𝑥 is. So we can work out what 𝐿 is. Let’s rearrange this equation to solve for 𝐿. We can subtract 𝑥 from both sides of the equation, which leaves us with 𝑠 minus 𝑥 is equal to 𝐿. At this point, all we have to do is to plug in the numbers. We know that 𝑠 is 0.48 meters and 𝑥 is 0.05 meters. And so we find the value of 𝐿 that’s 0.43 meters. And this is our final answer.
Know that the value 0.43 can be a bit awkward to use sometimes because it’s got decimal places in it. And we can, if we want to, write this length in centimeters rather than in meters. Now we can recall that the conversion is that one meter is equal to 100 centimeters. And so 0.43 meters is equal to 0.43 times 100 centimeters, which ends up being 43 centimeters.
And this is a much nicer value to use because it’s a whole number. So we can also give our final answer in centimeters. Of course both of these values are correct and valid. So at this point, it’s just a matter of personal choice. So anyway, our final answer is that the length of the spring when it’s not extended is 0.43 meters or 43 centimeters.