section 7.4 worked on by a spring force so let's spring into action what is a spring force a spring force is a variable amount of force that comes from a spring right so a tightly round coil a really loose example is like a slinky but you see springs in shock absorbers in cars for instance they pop up all over the place i don't have one in my home uh but the closest example i have is like a rubber band right so it has a very particular mathematical form and many forces in nature have this form so a rubber band is a sort of parallel even though it's not exactly the same so a spring or rubber band has a relaxed state a default state where it's not stretched or compressed right it's just able to be doing its thing and that's what's shown in this picture where the spring is there at its natural length there's a block attached to it but it'll just sit there for all of time but if we then pull the block out or stretch the rubber band out now it's no longer in a relaxed state right the spring exerts a restoring force that attempts to return it back to that relaxed state so you feel it when you stretch a rubber band that it doesn't want to stay stretched right it would much rather be in that simple relaxed state it's the same thing with the spring and the difference with a spring and a rubber band is the the spring that's true also if you compress the spring right if you push it together it will also exert a force that wants to get it back to the relaxed state the ribbond if you push it together it doesn't have much of a force trying to restore it to the equilibrium state so that's where the analogy breaks down a bit but for the stretching you can certainly see the parallels there so this is known as a restoring force it's a very particular one and the restoring force uh is the spring force that's known as hooke's law it says that the force of the spring is equal to negative k which we'll get to in a moment that's the spring constant times d the displacement and the negative sign here is that the spring force always opposes the displacement right so if i displace my rubber band up then my thumb feels a force pulling down right so if i've stretched it up the force is trying to pull it back down to that equilibrium state spring constant k is a measure of the stiffness of the spring or you can think of it as the springiness of the spring so a really thick rubber band or thick spring is going to be have a large cave value where it's really stiff it's hard to pull it apart if you have a small spring constant it's a sign that's a really loose spring so a slinky is a spring with a very small spring constant very low okay if you pull on the slinky there's not a huge amount of force pulling it back together note an interesting thing with the spring force is it's a variable force it's a function of position it exhibits a linear relationship between the force and the displacement what does that mean well that means when you pull the spring out of place the farther you pull it out of its relaxed state the larger the force gets versus when you pull it just a little not very much force there if we're just looking at spring along the x-axis horizontal then we can write that the spring force is equal to negative kx and we'll often use this because we'll often look at a spring in just a single dimension so more often than negative kd we're very frequently going to use negative kx as our spring force so that's worth noting now we said that we're going to find the work done by the spring force well if we take the force along the x-axis we can find the work by integrating we can't just plug this force into the equation we already had of the fd cosine phi why not well the problem there is that that only works for constant forces this force is varying depending on the position or the displacement of the spring so we have to do a full integral so if we integrate the work is the integral of the force from the initial to the final and we can plug in our negative kx there right so we would get that the work is equal to this uh integral of negative kx dx right and the k is a constant so that could come outside of the integral along with the negative so then we're just integrating x dx what's the integral of x dx it's x squared over 2 right so if we wrote this out we would have negative k out in front times x squared divided by two and that's going to be from the initial to the final so if we plug this in we'd have x final squared over two minus x initial squared over two but there's that negative sign out front still because that negative sign it flips the order around and so we find that the work done by the spring is what positive one half k x initial squared minus one half k x final squared so this is a handy equation you don't have to re-derive it for the most part you can just take this end result if you have a spring this is the amount of work done note here that the work here can be positive or negative it'll just depend on the net energy transfer right so if it let's see if the initial here we have it if the the work is positive the block ends up closer to the relaxed position closer to x equals zero than it was initially that means we're going the direction the spring wants to go the work is negative if the block ends up farther away from x equals zero and it's zero if the block ends up the same distance from x equals zero and you can see that if you plug in some numbers we'll see an example of that in just a little bit before we get to that we can note that we can simplify the equation if we just say that our initial position is the relaxed position that we're starting in the center then the work done by the spring is negative one half k x squared so this is often a handy one that we can just assume we started at zero and any displacement from the relaxed state the spring is doing negative work if there's an applied force like if we're carrying about the work done by our hand and stretching the spring or compressing the spring then we can look at the case where the initial and final kinetic energies are zero and once again when we say hey we have delta k is equal to the net work done if delta k is equal to zero then the total sum of the work has to be equal to zero so the work by the applied force plus the work done by the spring and so we see that the work done by the applied force has the exact opposite sign as the work done by the spring equal and opposite so the work done by the applied force displacement is the negative of the work done on it by the spring force all right so let's get to that example as i mentioned this is a checkpoint for the three situations the initial and final positions respectively along the x-axis for the block and the figure are negative three centimeters two centimeters negative two centimeters two centimeters and two centimeters three centimeters in each situation is the work done by the spring force on the block positive negative or zero so i want you to think about this come up with an answer to parts a b and c before you move on so go ahead and pause the video now all right so let's take a look the key is that we have initial and final positions right x i and x f in each of these pairs and our equation for work we know is going to be the x initial squared minus the x final squared so if we pull out the one half k in front and we'd see that the interesting part of what's changing is the spring staying the same here is the x i squared minus the x f squared note this feels very backwards to how we're used to we typically have final minus initial but that negative sign just flipped everything around here so what does that mean well that means we can plug these numbers in for each of the pairs to see what happens right x initial is negative 3 centimeters squared and then that's minus x final two centimeters squared so if we have negative three squared that's gonna come to positive nine minus four so that's gonna be positive five so that comes out to a positive number does that make sense conceptually well we've gone from a magnitude of three centimeters away to a magnitude of two centimeters away we've also crossed the relaxed state in the process we've ended up closer to the relaxed state so that should be positive work done by the spring so this is looking good let's now look at this next case negative two centimeters to two centimeters well for squaring it it's just gonna be two four minus four that's gonna be zero so for the second case spring is not doing any work net right it was doing some positive work to get to the relaxed state perhaps and then doing negative work to pass by the relaxed state but the net work comes out to zero then finally case c we go from two centimeters to three centimeters so we're ending up farther from the left relaxed state so that sounds like it's going to be negative work because we're getting the spring is going the way it does not want to go and sure enough we would have uh four minus 9 2 squared minus 3 squared so we would get a negative out of that so we can just double check our answers oh and they aren't listed here but it is indeed positive and zero and negative so there you have it this introduces the idea of a spring force in the first place and how we can calculate the work done by a spring force