in this video we're going to talk about hooke's law and springs we're also going to go over some problems associated with it so what's the basic idea behind hooke's law so let's say if we have a horizontal spring and so that's the natural left of the horizontal spring now what we're going to do is we're going to apply a force to stretch the spring towards the right we're going to call that force fp since we're pulling the string towards the right now that force is positive the displacement of the spring is also positive now we stretched it a distance which we'll call x it turns out that this force the force that's needed to stretch it by distance x is proportional to x so f p is equal to kx where k is a proportionality constant also known as the spring constant now this equation works up to a limit so f and x are proportional up to a limit once you pass the elasticity region then the spring can elongate without much extra force so we're going to focus on the elastic region where f and x are proportional now turns out that as you stretch the spring once you're at this point and you apply a force fp if you don't increase the force it's not going to stretch further so right now it's an equilibrium which means that there's another force that is pulling the spring back to its original length and that force is known as the restoring force which you can call it fs or fr and the restoring force is in the negative x direction now these two forces they're equal in magnitude but opposite in direction because once you apply a force fp the springs at rest it's not moving to the left or to the right once you let go fs takes over and it's going to cause a spring to snap back to its original length so because fs is a negative we can say that the restoring force is negative kx and this equation is associated with hooke's law so the magnitude of the restoring force is proportional to how much you stretch or compress the spring from its natural length so now that you've received a basic introduction into hook salon let's go ahead and focus on finishing this problem so number one a force of 200 newtons stretches the spring by 4 meters what is the value of the spring constant so if the force that we use to stretch it which is f p is equal to kx then the spring constant k is simply the ratio of the force and the distance that you stretch the spring so we have a force of 200 newtons and it stretches the spring by distance of 4 meters so 200 divided by 4 is 50 and we can see the spring cost that has unit newtons per meter so it's 50 newtons per meter so what this means is that in order to stretch the spring by a distance of 1 meter a force of 50 newtons is required in order to stretch it by 2 meters you need a force of 100 newtons to do so and so the spring constant it tells you how much force you need to stir uh excuse me stretch the spring by one meter so let's say for example if the spring constant was 300 newtons per meter that means a force of 300 newtons is required to stretch a spring by one meter now looking at these two spring constants which one is more stiff the spring that's 300 newtons per meter or 50 newtons per meter this spring is going to be more stiff it's going to be hard to stretch or compress the spring because it has a higher spring constant this spring is going to be loose it's one of those light springs that you can easily pull apart so as k increases as the spring constant increases in value the stiffness of the spring increases as well let's move on to number two the spring constant is 300 newtons per meter what force is required to stretch a spring by 45 centimeters so let's use this equation fp is equal to kx so our goal is to find fp k is 300 newtons per meter now we don't have x in meters we have it in centimeters so we need to convert that to meters in order to convert centimeters to meters you need to divide by a hundred one meter is 100 centimeters and so these units will cancel 45 divided by 100 is 0.45 meters so if we multiply the spring constant by 0.45 meters we can see that this is going to give us the unit newtons which is the unit of force so it's 300 times 0.45 and so that's going to be 135 newtons so that's the force that's required to stretch the spring by 45 centimeters keep in mind if the spring constant is 300 newtons per meter in order to stretch it by one meter it requires a force of 300 newtons so thus to stretch it by half a meter would be 150 newtons so 0.45 meters which is just under a half would make sense that it's 135 newtons it's always good to take a mental check of your answers so let's make a table between x and the force applied so the spring constant was 300 newtons per meter so that means at a distance of 1 meter a force of 300 newtons is required so if we divide the distance by 2 we should divide the force by 2. so therefore our answer at 0.45 meters is 135 which makes sense so this is less than 25 and this is less than 150 so always check to see if your answer is reasonable does it make sense now let's move on to number three a force of 250 newtons is required to stretch a spring by 24 centimeters how far can the force of nine hydrogen stretch so we're given a force and a distance and we're given another force and we're trying to find another distance you could find a spring constant k or you could come up with another equation to do this so i'm going to show you two ways of getting the answer the first is to calculate the spring constant k it's going to be the force divided by the distance so a force of 250 newtons is required to stretch it by 24 centimeters which is 0.24 meters if you divide it by 100 so 250 divided by 0.24 will give us a spring constant of 1041.6 repeating now to find out how far it's going to stretch we need to calculate the value of x and x is f divided by k so we have a new force of 900 and we're going to divide it by 1041.6 repeating so this is going to be about 0.864 meters which in centimeters is 86.4 centimeters if you multiply by 100 so that's one way in which you can get the answer now let me show you another way let's take the second force and let's divide it by the first force the second force is k times the second distance and the first force is k times the first distance so because k is the same we can cancel it so therefore the ratio of the forces is equal to the ratio of the displacements in x and it makes sense because x and f are proportional to each other so we're going to call this f1 and this is going to be x1 and this is f2 and we're looking for x2 now the unit centimeters will cancel so it doesn't matter if you use meters or centimeters when dealing with ratios the ratio will still be the same so let's replace f2 with 900 newtons and f1 with 250 newtons our goal is to calculate x2 and x1 is 24 centimeters so let's cross multiply so first we have 900 times 24 so that's 21 600 with the units newtons times centimeters and that's equal to 250 newtons times x2 now in order to get x2 we got to divide both sides by 250 newtons on the right side these two will cancel giving us x2 on the left side notice that the units newtons will cancel leaving behind centimeters which is going to be the unit for x2 so it's 21 600 divided by 250 and so you're going to get 86.4 centimeters so you have two ways in which you can get the same answer consider this problem how much work is required to stretch the spring by 75 centimeters and we're given the spring constant so let's draw a spring now the force that's required to stretch the spring above its natural length let's say this is the natural length of the spring is we're going to call it fp now fp is equal to kx now once you stretch it and you stop stretching it you still need to apply a force to hold its position now if it's at rest that means that there's an equal and opposite force that wants to snap back the spring back to its natural length and that is the restoring force now this force is positive it's going in the positive x direction and this force is negative so according to hooke's law the restoring force is negative kx but we're going to focus on this force how much work is done by this force in order to stretch the spring by a displacement x x is the distance between the natural life of the spring and its current life so it's how much it stretches or compresses by now work is equal to force times displacement so the displacement is x so work is equal to f x now f is not a constant value f depends on x f is a function of x so if we were to make a graph between force and displacement it would look something like this now f is not a constant value but it's a function of x it's equal to k x and so all we have is a linear equation where the slope is k so this is going to be a straight line that looks like this now whenever you have a force displacement graph we know that work is force times displacement but for such a graph the work is equal to the area under the curve so what is the area of the triangle the area of a triangle is one half base times height in this example the base of the triangle is equal to x and the height of the triangle is equal to f so therefore the area is one half f times x where f is the maximum force value at a distance x so this equation holds true if the force is constant so if he had a graph where this is f and this is x and the force was constant the area will be the area of the rectangle which is left times width that will be simply f times x there won't be a half in front of it that's if the force is constant now what we have is the variable force that increases according to the displacement or how much you stretch it so therefore the work done by that variable force is the area under the curve which is one half f times x so the work required to stretch the spring is one half times the force the maximum force times x and based on hooke's law we know that f is equal to kx so what i'm going to do is replace f with kx and keep in mind this x is really a delta x so this is going to be k delta x and this is supposed to be delta x because it's really the change in the position so let's say if this is the natural left of the spring it's at position xa and you stretch it to a new position xb this is the difference between xb and xa so technically it's delta x so therefore the work required to stretch a spring is going to be one half k times delta x squared so this is the equation that you want to use so k is 250 newtons per meter and we want to stretch the spring by 75 centimeters however we need to convert that to meters to convert centimeters to meters divide by a hundred so this is going to be point seven five meters squared so it's point seventy five squared times two fifty times point five so therefore the work required is 70.3 joules of energy now what is the elastic potential energy stored in a spring when it is stretched 25 centimeters by a 450 newton force so let's come up with the equation to calculate the elastic potential energy first so we said that the work required to stretch a spring by the applied force is one half k delta x squared now the work done by such a force is equal to the change in the potential energy if the acceleration is zero which it is the applied force and restoring force they're equal to each other so work is equal to the final potential energy minus the initial potential energy now some textbooks may describe the elastic potential energy as us capital u can be described as potential energy so if you see us that's elastic potential energy ug gravitational potential energy so just keep that in mind i like to use pe because p e potential energy it makes sense so what i'm going to do is replace the w with this expression so one half k delta x squared is equal to the stuff that we have on the right and delta x is basically the final position minus initial position so we have one half k x final squared minus x initial squared so one half k x final squared minus one half k x initial squared is equal to the final potential energy minus the initial potential energy so what does this all mean well we can see that this is equal to the final potential energy and this is equal to initial potential energy therefore the elastic potential energy in general is simply one half kx squared so notice the difference between this equation and this equation so what do you notice the work depends on the change in position whereas the potential energy it depends on x and x by itself it's has to be relative to the natural length of the spring so the potential energy is based on the relative length between where you are and the natural length of the spring the work required doesn't have to be associated with the natural length for the spring it's simply the energy required to move it from position a to position b for this equation position a has to be the natural length of the spring but for this one position a or b does not have to be the natural left of the spring so keep that in mind when i mean natural benefit you can also describe it as the natural position of the spring which is usually defined as x equals zero so let's focus on this equation what is the elastic potential energy stored in this spring so we're given x and we're given the force f so we know that the elastic potential energy is one half k x squared the only thing we're missing is k so we could find k using this equation f equals k x so k is the ratio between the force and the displacement relative to the the origin or the natural position of the spring so it's going to be 450 newtons divided by 25 centimeters in meters which is 0.25 if you divided by 100 so 450 divided by 0.25 is 1800 so that's the spring constant so this spring is very stiff as k increases the stiffness of the spring increases so just to stretch the spring by distance of one meter requires a force of 1800 that's what the spring constant tells us so now that we have k let's calculate the potential energy so it's one half times k times x squared where x is 0.25 meters so .25 squared times 1800 times 0.5 that's equal to 56.25 joules and so that's the answer how much work is required to stretch a spring from 4.5 centimeters to 8.2 centimeters given a spring constant of 150 newtons per meter so let me give you a visual first of what we have here so let's say this is the spring at its natural length so at this point x is equal to zero now the spring is currently stretched to this position let's call it xa that's when x is equal to 4.5 and we need to calculate how much work is required to stretch it further to position b where x is 8.2 centimeters so how can we find the work required to go from position a to position b go ahead and try that problem the equation that you need is this one one half kx i mean k times delta x squared now you need to be careful because there's something you don't want to get mixed up with when i wrote this expression i meant that delta x squared is equal to one half k times x final squared minus x initial squared so not like this not one half k x final minus x initial squared if you do it this way you will get the wrong answer so it's the difference of the squares of x not the square difference of the x values if that makes sense so this is the right way to calculate it because then work is equal to the change in potential energy so if you do it this way you're not going to get the right answer i just want to highlight that point so it's going to be one-half times the spring constant which is 150 times the square of the final position which we need to be in meters so that's going to be 0.082 meters squared minus the square of the initial position which is 0.045 meters squared so go ahead and type this in exactly the way you see it so you should get .352j for the work required to stretch it from position a to position b now for those of you who may want another way to verify this answer calculate the potential energy at points a and b so the potential energy at point a is going to be one half k x squared where x is that position a so it's one half times 150 times 0.045 meters squared so at position a the elastic potential energy is 0.1519 joules now let's calculate the elastic potential energy at position b so it's one half k xb squared so that's one half times 150 times .082 squared and so this will give you 0.5043 joules so keep in mind you can always calculate the work by calculating the change of potential energy so it's going to be the difference between these two values so it's 0.5043 minus 0.1519 and so you get the same answer 0.352 joules so the work done by a force is equal to the change in potential energy if the net acceleration of the system is zero now there are a few additional details that i want to mention regarding the work required to pull a spring as i mentioned earlier in the video fp is the pulling force or the force that we use to stretch a spring and based on the way the the drawn that we have on the screen since it's directed to the right this is going to be defined as a positive force and we described fs as the force exerted by the spring which is a restoring force that force tries to bring the spring back to equilibrium as we stretch the spring to the right the displacement vector is towards the right and we know that work is force times displacement notice that these two vectors are in the same direction therefore the work that's required to stretch a spring is positive now the work that's done by the spring itself as it pulls the spring back to equilibrium that work is negative as you can see the force and the displacement vectors they're in opposite directions so therefore the work required to stretch a spring is positive one half k delta x squared now the work done by the elastic force that is fs the work done by the spring that's going to be negative one half k delta x squared as the spring is being pulled to the right now this makes sense because in other videos i mentioned that the work done by conservative forces like gravity the electric force or in this case the elastic force negative fs that work is negative times the change in potential energy and the change in potential energy is equal to this expression it's one half k delta x squared this is for those of you who may be wondering whether or not you should have a positive or negative sign it really depends on which force you're talking about and where the spring is being whether it's being stretched or compressed so just remember the work is positive if the force and the displacement vector are in the same direction but it's negative if they're in opposite directions you