A spring with a force constant of 64 newtons per meter is pulled by a force of 4.8 newtons. By how many centimeters does the length of the spring increase?
So in this question, we’ve got a spring which first of all has a force constant of 64 newtons per meter. Let’s draw a diagram to illustrate this. So here’s the spring in its natural length before any forces have been applied to it. And we’ve been told that the force constant is 64 newtons per metre. Let’s label this quantity 𝑘. And we know that 𝑘 is equal to 64 newtons per meter. Now what happens to the spring is that a force of 4.8 newtons is applied to it and this results in the spring extending. Now when a spring extends, it exerts its own force in the opposite direction.
In other words, when you exert a force to the right on the spring as we’ve drawn it, the spring itself exerts a force to the left. Now the spring will reach equilibrium when these two forces — the force to the right and the force to the left — are balanced. In other words, the force exerted by the spring to the left must also be 4.8 newtons and it’s at this point that the spring stops extending.
Now there’s a relationship that we can recall that links together the force exerted by a spring, the force constant of the spring, and the extension of the spring, which we’ll call 𝑥. That relationship is that the force exerted by the spring is equal to the force constant multiplied by the extension. And it’s really important to remember that it’s the extension that counts. It’s not the length of the original spring or the length of the stretched string.
Luckily for us however, the question actually asks us “by how many centimeters does the length of the spring increase?” Well, the length of the spring increases by 𝑥. So we’re trying to find out 𝑥. And even more luckily for us, we already know the values of 𝐹 and 𝑘. So we can rearrange this equation a little bit to find out what 𝑥 is. We divide both sides of the equation by 𝑘. So we’re left with 𝐹 divided by 𝑘 is equal to 𝑥. Now, we can just plug in the values of 𝐹 and 𝑘 so that we have 𝑥 is equal to the force of 4.8 newtons divided by the force constant 64 newtons per meter.
Now because each of these quantities is in its standard unit, the standard unit of force is newtons and the standard unit of force constant is newtons per meter, we’re going to find the extension 𝑥 in its standard unit of meters. So when we evaluate the right-hand side of this equation, we find that 𝑥 is equal to 0.075 meters.
However, the question asked us by how many centimeters does the length of the spring increase. So we need to convert 𝑥 from meters to centimeters. To do this, we’ll recall that one meter has 100 centimeters in it. So to go from a quantity in meters to a quantity in centimeters, we just need to multiply by 100 and change the unit from meters to centimeters. Doing this gives us a value of 7.5 centimeters. And that is our final answer.
The length of the spring increases by 7.5 centimeters.