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Hooke's Law

Teacher/Instructor Jonathan Osbourne
Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

Hooke's Law is used to determined the restorative force or the amount of elasticity. Larger distortion would result in a larger force. Hooke's Law is expressed in the equation F = - k(x), in which k is the spring constant and x is the displacement.

So let's talk about Hooke's law. Hooke's law has to do with springs. Alright. Well, what's the deal with springs?

Well, springs have a restorative force. So that means that if I pull on them, they try to pull back. They want to be a certain length and it's called their relax length. So if you pull on them they're going to pull back and that's because they're springy, right? So a larger distortion, the more that stretch it the more force that we need to do that.

Now the force that we're going to pull with is the same as the force that the spring is going to pull with when we've got it at rest because then it's in equilibrium. So I pull it, and I keep it there, the force that I'm applying is equal to an opposite to the force that the spring is applying. So Hooke's law says that that law is proportional to how much I stretch the spring. Alright.

So f=kx. x is the length of the spring now minus its length when it's relaxed and nobody's pulling on it. k is a constant called the spring constant. And it just depends on which spring we're talking about. If k is really really really big, then that means that the spring is very rigid so you should think like car shocks or those springs that, that hold up like a swing outdoors.

A really really really small spring constant means that the spring is really loose, really easy to pull on. You should think like a slinky or something like that. Alright. So let's do some problems. First off, the direction decreases x, it acts to make x smaller because we don't want any distortion at all. It wants to be its relaxed length. So for that reason f the force is negative kx. It's the opposite of k times x. Alright. The magnitude is just proportional but the direction is opposite because it wants x to be smaller. Alright.

So let's do an example. A mass m stretches a spring a distance x. So you should think a spring like this hanging, I hang a mass m on it and it stretches it a distance x and I want to know how far will 3m stretch it? Alright. Now, we don't have any numbers in this problem. But we don't really need them. Look, 3m is going to give a force it's weight of 3 times as much as m. So if the force is 3 times as much, the spring constant is constant, it means x got to be 3 times as much. Easy enough. Alright. Let's go to the next one.

This one's got some numbers in it. A 200 gram mass, stretches a spring 50 centimeters. What is the spring constant? The important thi- there's a couple of important things here. So first of, we're not in SI units. So we got to change to SI units if we want everything to work nicely. So that's the first thing. Second thing, we're going to write down f=kx. I don't care about the direction so I'm not going to write the minus sign. So what's the force? The force is not 200 grams. The force is mg. You got to multiply by the acceleration due to gravity. So we'll have m 0.2 g, I'm just going to use 10. It's 9.8 but whatever. You're going to use the calculator, use 9.8. I'm not going to so 10. Alright. Equals m and I've got k and then x, just like that.

Now, 0.2 times 10 is 2, right? 2 divided by 0.5, dividing by 0.5 is like multiplying by 2. 2 times 2 is 4 and now all I need is a unit. Remember when you're writing your answer you always have to include the unit. Well, what's the unit? Well, f=kx. f is a force so it's Newtons, x is a length so it's meters. So k must be Newtons per meter. And there you go. Alright. Let's go ahead to the last one.

How far will a force of 6 times 10 to the 3 Newtons, stretch a spring with a spring constant of 3 times 10 to the 4 Newtons per meter. Alright. Again, f=kx. Now here I want x. How far, alright? So, I'm going to solve for x. x=f over k and then i just plug in. So it will be 6 times 10 to the 3 over 3 times 10 to the 4. Alright.

Now, as with any time you've got scientific notation, you always do the numbers first and then the tens. 6 divided by 3, 2. 10 to the third divided by 10 to the fourth, that's 10 to the 3-4. So it's 10 to the -1. What's the unit? It's x. Everything's in SI units, so everything's in SI units and now if we wanted to be all cool about it we could write it as 20 centimeters. But 2 times 10 to the -1 meters is fine too.

And that's Hooke's law.