**Example:** Energy in Horizontal Springs

Hey guys, so in this video I want to talk about energy problems involving springs let's check out, so we're first going to look into energy problems with the horizontal springs but before I talk about energy let me remind you that when we have problems involving springs and stationary objects we solve these using forces, F=MA and also the spring force, let me remind you about the spring force if I push against a spring like this in such a way that the spring compresses a distance X the force that you apply or that I apply a F (you) has the same magnitude as the force of the spring except that the spring will be pushing back this way so this is the force applied by you and the spring will push back this way and this equation here it's called Hooke's law, there's a negative sign here remember what the negative tells is that if you push this way so the spring moves this way the spring will push back the opposite direction so that negative just tells you that the force of the spring is opposite to the way that you compress the spring or if you stretch it out the other way, it's opposite to the deformation that's all that negative does and for practical purposes I like to ignore that negative because it's just telling me conceptually that there's a difference in direction, so I like to think of it as this that your force is the same as the force of spring and that equals Kx where X is the deformation and K remember is the spring force constants or the spring coefficient it's a measurement of how strong the spring is, cool. Now that's with a stationary object with objects when you have springs and moving objects we're going to use energy instead we can't use forces we have to use energy and then the last point I want to mention here before we do an example is that springs are always going to be massless at least in basic physics springs will be considered masses to make things simpler and because of that they won't have kinetic energy or gravitational potential energy and you can look at the equation and see this right away, kinetic energy requires a mass springs massless, potential energy or gravitational potential requires a mass but springs will be light and therefore it is considered massless elastic energy, Elastic energy is 1/2Kx squared so springs will only have an elastic energy, so the only time we're going to have kinetic energy or gravitational potential energy is if there's a block which has a mass that is touching the spring, OK? That's one example here than we got a practice problem and let's knock this out. So, I have here a 4 Kg block so mass=4 is in a smooth horizontal surface so it looks horizontal there, smooth no friction, you push the block against the horizontal spring that is attached to a wall with a constant horizontal force so you're going to push against the block like this, right? So this is you pushing against the block and then obviously the block is going to....The spring is going to push back against you, you push against the block and when this force is 100 Newtons So when this force here is 100 Newtons which means this force is also going to be 100 because of action reaction this spring would have compressed a distance of 20 cm, so here's the idea let's say it starts here and then you push all the way to here when this force at this point right here is 100 it means you will have compressed the distance D or X of 0.2 meters, this is 100 Newtons, OK? And then when you release the block obviously if you compress the spring this way when you release what is it going to do? The spring is going to extend back, spring are always trying to go back to their 0 to their equilibrium position the point where they're uncompressed or not deformed, right? They are at their relaxed natural state so the spring is going to push back and so you're going to release here and then the spring is going to push back this way, OK? So, this is going to be our initial position and our initial compression is 0.2 because I have compressed 0.2 and my initial velocity or speed is 0 because I'm going to compress hold it there and then release from rest and it's going to push out this way. When you're back here, your final compression will be 0 and then if you read the question eventually we're going to ask what is the block's launch speed? So, we want to know what is the final velocity? Launch speed is the velocity with which the spring launches the block obviously and that happens when the spring is back to its equilibrium and that's because at that point the spring will stop and the block keeps going. Alright so let's see, first question here is actually what is the spring's force constant or spring coefficient or the stiffness of the spring? It's K so let's try to find K, the only.....There are two equations where K shows up one of them is this one here the force F=Kx or this one here the potential energy, so I'm going to have to use one of these two equations to find K and if you look at all the information you're given you're given the force and the compression since those are the only two pieces of information that I have it should be pretty obvious that I should be using the F=Kx equation, so once again the force of you has the same magnitude as the force of the spring which is Kx, your force is one hundred I'm looking for a K, X is 0.2, so when I hold it at 0.2 it requires a force of 100 so K is 100/0.2, K is 500 Newtons per meter all the units are....We're using all the right units so you get the units of Newton per meter, that's part A.

For Part B, make a little space here, for Part B we're being asked what is the maximum potential energy? Now potential energy here, there's no gravitational potential because we're on the floor level so potential energy here means the maximum elastic potential energy, now elastic potential energy is 1/2Kx squared, K is a constant it's not going to change but if I want the greatest possible value of you I need to have the greatest possible value of X because the other numbers are constant, if I want to maximize one I have to maximize the variable on the other side and the maximum compression you can have in this situation the maximum compression that you do have is the 0.2, alright? So, it's basically asking what is the potential energy right here when the spring is fully compressed or not compressed but compressed the most, right? Maybe could have been compressed a little bit more, anyway so let's plug in these numbers 1/2, K is 500, X is 0.2 squared, and if you plug all this I have it here you're going to get 10 joules, so K is 500, potential energy max is 10 joules let me put a little max here.

Now for Part C, so just real quick here I had to use this equation because it's the only one that had all the variables I needed, this was straightforward I was asking for the potential energy so you plug all the numbers you have into the potential energy equation, now here this problem is talking about two points, it's saying if I release here what's the speed over here? So, I'm going to use energy, right? Because I have a moving spring so I'm going to use energy so kinetic initial + potential initial + work non-conservative is kinetic final + potential final, there's no kinetic energy to begin with because you push, hold it there and then you release it from rest, there is potential energy (elastic potential energy) because a spring is compressed as long as you have an X then you have a potential energy, there's no work done by non-conservative force, non-conservative forces are friction there's no friction here or external forces, a spring force is a conservative force so it doesn't count as non-conservative, there is going to be a kinetic final right here there is a velocity and that's actually what I want to calculate what is that velocity or speed? But there is no potential because at the end of this the launch speed happens when you've crossed 0 and yet at that point your potential energy is 0. If you look here what's happening is that all your potential energy which is the stored energy due to the fact that you compressed against the spring goes entirely into kinetic energy so your compression gets transferred into speed, right? So, let's do this 1/2Kx squared this is initial, I will put a little I there and this is 1/2MV squared and that's final so I'll put a little f there, notice that the masses won't cancel anytime you have a spring problem like these masses will not like we're used to seeing most of the time, OK? And if I want to know V final which is what we're looking for V final I can move the M to the other side and then I can take the square root of both sides at this point obviously you can just plug in a bunch of numbers but I do want to show that this X technically gets out of there, right? So, if you want to make this I guess nicer or neater probably not necessary unless your professor wants you to make it as cute as possible toward the end but you know this is the simplest form you can have now we can plug in some numbers here, K is 500 we solved it earlier, mass is 4 we were given that and the initial position we're given as well, that's 0.2 and if you do all this you get 2.24 meters per second, OK? And that's our final velocity which is our launch speed, Cool? So that's it for this one I have a practice problem for you guys that's basically the opposite instead of you launching a spring launching a block with a spring there's a block there will hit a spring and compress against it, I want to give this a shot and hopefully get it right let's try it.

**Problem:** A 4-kg block moving on a frictionless, horizontal surface with 20 m/s strikes a massless, horizontal spring of force constant 600 N/m. Calculate the maximum distance that the block will compress the spring by.

EXTRA: What maximum acceleration will the block experience? (hint: this happens when it temporarily stops).

7m **Example:** Springs in Rough Surfaces

Hey guys, in this video I want to talk about energy problems with springs and friction so we're going to have horizontal springs, a block attached to it or being pushed by it and we're going to have kinetic friction as well let's check it out, so what's special about these problems is that when we have springs horizontal springs and rough surfaces there are going to be two forces acting on the block, the spring will be pushing the block and there will be friction as well so we have two forces will be force of spring and friction and there will be kinetic friction, right? And there are two distances, there is going to be the distance that the block will travel we usually call this D or delta X and the compression of spring the deformation on the spring, compression it could also be extending the spring and deformation would be a more precise term there and that's usually we usually use X to denote the two sometimes those distances will be the same, let's check out this problem here.

I have a massless spring, remember springs are always massless so the word massless here doesn't really do anything it's always going to be that way with a force constant that is little k of 500 Newtons per meter those are the correct units so force constant is the spring coefficient and the spring stiffness and is set up parallel to a long horizontal surface so the spring is horizontal and therefore it is parallel to a surface, right? Parallel to a surface and it's attached to a vertical wall as shown so it's just as described in the picture, 3 kilograms block, mass=3 is pushed against the spring until it's compressed by 4 meters so you're going to get this block put it next to the spring and push until you've compressed 0.4 meters, so you're going to go from here push all the way until you're somewhere over here and we're going to say that this gap is 0.4 meters, it says that the coefficient of friction the block surface coefficient of friction in other words the coefficient of friction between the block and the surface right here is 0.6, that's mu, alright? And then it says when you release the block the spring pushes out accelerating the block makes sense, right? So, block's here you compress a spring when I release it shoots it out, now I want to calculate two things the speed with which the spring launches the block so the......We'll start here and then right here the block.... the spring stops at this point when it gets back to X=0 no compression, the spring stops but the block keeps going, this is the launch speed right here so I want to know what is the velocity at this point? But the block keeps going it says how far will the block travel after it leaves the spring, the block keeps going but eventually it stops, right? The reason it stops is because there's friction so it's going to be slowing it down, you can also tell that it stops because it says how far will the block travel? Ok calculate how far the block will travel it implies that eventually stops that there's a maximum distance, so there are two points and I'm going to call them A, B and C, and I want the velocity at point B and I also want to know this here, how far it travels after it leaves the spring? So, I'm going to call this D2, I'm going to call this right here D1 and D1 is 0.4, what else do I know? I know the velocity at point A is 0 and I know that the velocity at point C is 0 and that's because I get the block compress all over here it's not moving and then so this is A, Va=0 I release it starts moving, right? Gets to Vb keeps moving eventually stops at Vc, what else do I know? I know compressions, the compression at point B right here is Xb=0. At point A I have Xa=0.4 that's spring compression, right? And from here to here there's no compression so the spring has already stopped so Xc=0, OK? The spring decompresses from here to here and then it's just the block moving by itself, so I hope you see what's happening here in solving this problem especially after you get the hang of this you probably wouldn't take this long or analyze this much I just want to kind of give you a big overview of what's happening here so I want to move Vb and distance 2, you can also think of this as the distance between B and C, cool? So, Part A what is Vb? This is an energy problem because when we have moving springs it's always going to be an energy problem, I know information about A, I know the velocity at A is 0 and I know that compression and I want to know some information about B I want to know what is Vb, so I'm going to write an energy equation from A to B, that's our energy equation, kinetic energy at point A does not exist because there's no speed it says it right there, potential energy at point A there's no gravitational potential energy because everything is on the surface or on a horizontal surface so there will be no heights but there is elastic potential energy, work non-conservative does exist in this situation because remember work non-conservative is the work done by you which is an external force or the work done by friction and in this case there is no work done by you because as soon as you release you are not doing any more but there is friction and the work done by friction remember is -Fd because we're talking about kinetic friction, coo. There is kinetic energy at point B that's after you've moved, right? So, at point B you have a velocity so there is a kinetic energy, there is no potential at point B because at that point all your potential energy has become kinetic energy and also because the spring at that point is not compressed anymore it's back to X=0, alright? So let's write this out, Ua is going to be 1/2kXa squared this is negative friction distance and this is a 1/2MVb squared, that's what I'm looking for right there, so now it's just a matter of plugging in all the numbers and we have all the numbers 1/2(500) the compression is the same thing as D1 so its 0.4, friction we have to calculate friction distance so I'll get friction in a second but the distance here is 0.4 as well, notice that these numbers are the same because if it's moving from here to here it's a compression of 0.4 and it's a distance of 0.4 because it's decompressing it, the block is moving while the spring has been decompressed so let's calculate friction, friction is mu normal always and normal is usually MG and it is MG in this case because the only two forces on the Y axis are normal and MG so they must equal each other, OK? So, normal is MG, therefore friction is mu MG we can calculate that real quick, mu=0.6, mass=3, gravity I'm going to use 10 so this will be 18, alright? Cool so I'm going to put an 18 here, 1/2 (mass=3) Vb squared, so I got all the numbers now I just have to work out the algebra here and get Vb by itself, this is 100, this is 7.2, this is 1.5 Vb squared if you solve this whole thing you get that Vb is 7.87 meters per second, that's part A let's solve part B.

Part B is asking us it sketch it gets launched with a speed of 7.87, part B is how far will it move after it leaves the spring? I want to know D2, D2 is the distance between B and C so obviously I have to write an energy equation from B to C, Kb+Ub+work non-conservative=Kc+Uc, there is kinetic energy at point B because you just got shot off you got launched, there's no potential energy at point B because at that point of spring is back to 0 so it's already used up all of its potential energy, there is work non-conservative because it's the work done by friction because there's still friction for this piece here, there's no kinetic energy at the end because eventually you stop and we know the maximum distance so that implies the final speed has to be 0 and there is no potential energy at the end either because again the spring's already back to 0, so I have here 1/2MVb squared+ the work done by friction, work non-conservative is just the work done by friction which if you remember it's negative Fd that equals 0, I'm going to move this over to the other side 1/2MVb squared=Fd, OK? And what we want to know is we want to know this distance right here, now if you want you can keep expanding this and you can replace friction with mu MG, we talked about just here how friction was mu, if you do this again you can at this point have just plugged in the numbers, I know friction is 18 but I just want to show that if you went all the way through to up to this point here the masses would have cancelled, right? And some professors will prefer if you solve this with letters or sometimes they'll specifically ask you and not give you numbers and just solve it in letters so it's a good skill to have if your professor likes that, so let's find D, D therefore is the Vb squared/2 mu G, OK?And if you plug in the numbers, this is 7.87 mu is 0.6, gravity's 10, we've got everything it's going to be 5.16 meters that's how far this thing will move on a surface before it gets to rest before it just stops, right? By the way part B has nothing to do with springs, right? It's just a little extra thing, we've done something like this before it's a very very similar question the stopping distance and part A is really what deals with spring but just to show that you can get a little more of the R2 distances D1 and D2, I got a practice problem here that's very similar there is friction throughout and now you're going to be hitting the spring and stopping it, right? And we're going to use conservation energy as well let's give this a shot hopefully you get it.

**Problem:** A 4-kg block moving on a flat surface strikes a massless, horizontal spring of force constant 600 N/m with a 20 m/s. The block-surface coefficient of friction is 0.5. Calculate the maximum compression that the spring will experience.

**Example:** Energy in Vertical Springs

Hey guys we're now going to solve energy problems with vertical springs let's check it out, so when you have a mass a moving mass that is interacting with the vertical spring means it's pushing against the vertical spring or being pushed by the spring something like this over here where a moving mass is going to push against the spring we have all three types of mechanical energy, we have kinetic energy, we have gravitational potential energy of the block or the mass that's because the mass has velocity it's moving up and down and the mass has potential energy the potential is changing because the heights are changing and they're also have the elastic potential energy of the spring, cool? So let's check this out here, I have a 5 Kg mass, so I will put here mass=5 is released from 3 meters above the top of the spring so this gap here is 3 meters, I'm going to leave some space because I want to call this point A and I want to call this point B, right here the beginning and the very top of the spring, the spring is a 400 meter spring this is the coefficient of the spring force coefficient and it says calculate the maximum compression that the spring will experience? So, if this falls from here to here it's going to hit against this little platform here and it's going to cause the spring to be compressed further down until the system stops, right? So, it's going to go down a little bit more to another point let's call this point C and it will have some compression, that maximum compression is what we want so I'm going to call this Xc because it's the compression at point C right here, OK? Now maximum compression is when it goes all the way down which means it implies that the kinetic energy here is 0 and that's because obviously the velocity at that point 0 maximum is it went all the way and now it stopped, OK? So how are we going to do this? We're going to do this using obviously the energy equation, now the energy equation is always from 2 points and you have to pick the 2 points you know information about A and you want information about C because you want to know what is this guy here, so we're going to write an energy equation from A to C to try to figure out what is Xc? So, I'm going to write Ka+Ua+work non-conservative=Kc+Uc. The block is released presumably from rest you can assume that so the kinetic energy begin at 0 there's no speed, potential energy remember there's two types of potential energy I'm going to split them up here there's gravitational potential energy at point A and elastic potential energy, well I mean there could be these two types of energies in this case you have a height but you don't have spring compression so the only one you have is gravitational, work done by non-conservative forces if you remember is the work done by you, you're not doing anything or no external forces you're just watching plus the work done by friction there's no reference to friction so we're going to assume that there is no friction, this whole thing is gone and on the right side you have no kinetic energy because at the end it stops, right? That's why it's maximum and you do have to have potential, now remember potential is there are two types you have at point C gravitational potential energy and at point C elastic potential energy at the bottom we're going to say that since this is the lowest point the height here is 0 therefore there is no gravitational potential energy but the spring is certainly compressed so there is some spring energy, Ok? So I have here MGH at point A and then I have here 1/2Kx squared at point C those are the two types of energies, I know all of these numbers except Xc well actually it's a little more complicated and I'll show you so let's go through this real quick, the mass is five that's easy, gravity I'm going to use a 10 now the height is a little tricky because the height is the distance from A all the way to C and that height is actually 3+X you don't know what X is but that's ok we can just put it here, 3+Xc, so let's see where this goes, I have 1/2(K=400) and Xc squared so notice that I have Xc twice and that's OK because it's just one unknown so we can solve this we're just going to make this look a bit cleaner so I have here 50x(3+xc) and over here I have 200Xc squared, I have to distribute this 50 here so it's going to be 150+50Xc equals 200Xc squared and you might notice what's happening here, I have one turn with an X squared 1 term with an X and one term with no X whatsoever this is a set up for a quadratic equation so unfortunately we have to solve a quadratic equation here, I have to get this in quadratic form and remember the quadratic form is AX squared +or- BX +or- C=0, if you need to review this please do remember that A has to be positive which means I have to move everything to the other side so that these two terms here the 150 and 50 are hanging out on the right side here with the X so it's going to be 200Xc squared-50XC-150, the physics is over here we just have to do the algebra so one of things you can do to simplify this is you can divide everything by 50 and then you get 4XC squared that's your A, -1 there's a 1 implied in front of the X here that's your B actually your B includes the minus so be careful there and then this is a -3 this is your C=0, So now it's write the quadratic equation Xc=(-b +or- the square root of B squared-4ac)/2A, I'm not going write the whole thing because I'm running out of space but obviously you have to know that, so -B, B=-1 all of that +or- B squared...I'm sorry the square root of B squared +4 sorry -4AC when you solve this the stuff that's inside of the square root that actually give you 49 which the square root of 49 is 7/2A which is 2x4 so this becomes a plus 1 or a 1 +or- 7/8, the +or- means we're going to do this twice, the first version is Xc=1-7/8 this is going to give you a negative number and a negative compression makes no sense so remember usually in physics when you do the quadratic equation you end up with one of the numbers being tossed out, one of the answers being tossed out the negative one so the other one is going to be positive 1+7/8, that's 8/8 and that's 1 meter and that's the final answer, so the total compression the maximum compression is exactly 1 meter, cool? Let's do the other one now.

**Example:** Energy in Vertical Springs

Alright, so here's another example and this is a pretty long one so you should try to write small to make it all fit, I have a vertical spring of force constant 800 that's the spring coefficient, K put it right there 800 and it is attached.... So the spring is vertical but it's attached to a horizontal surface as shown below, so this is as described in the picture there's a 10 Kg block that is placed on top of the spring just as shown and then it's allowed to descend slowly compressing the spring until it reaches equilibrium, allowed to descend slowly means that you're not going to just draw or put the spring or block here and release and let it fall because if you do that it's going to fall past the equilibrium point and it's going to start oscillating like this, OK? So, what you do is you put the block on top and you kind of hold onto the block and let it slowly fall, it's like a controlled fall so that it doesn't oscillate, cool? So that's how that works it's different from the other example where it was sort of an uncontrolled fall just hit the thing and it went all the way back down and then it would shoot the block back up, here it's a controlled fall which means that it will go down until it reaches equilibrium so it's going to start over here at point A and it's going to slowly descend until it reaches equilibrium at point B, so point B is equilibrium, OK? And the first question is what is the spring's compression X at equilibrium Xb? So, what is this compression here Xb, OK? Then the block is going to pulled down an additional 30 centimeters so you're going to pull the block down an additional 30 centimeters so this is 0.3 meters and then its release from rest. When it's released from rest what's going to happen just as an overview real quick is you're going to release it from here with an initial velocity of 0 which is the same thing as Vc=0 and it's going to go back to B Cross A and then go up a little bit more, OK? And the problems here parts B, C and D are asking what is the block's maximum speed? and when does that happen? I'll talk about that soon what is the block's speed once it's back to its original height, the original height is here and if I want to know the speed here I want to know what is Va? And then what is the block's maximum height? Maximum height is over here and I have A, B and C so then we're going to come back to C B and A I'm going to call this D Just because A B and C are taken so I want to know what is this height right here Hd? Now I notice that we want the height above the spring's original height so I want to know Hd from A not Hd all the way from the bottom, cool? So, it's a little bit different, alright? So, let's first look for Xb so I hope you remember what's special at equilibrium? When you have a spring a vertical spring at equilibrium the force are cancelling, that's the definition of equilibrium so at equilibrium sum of all forces=0 which means the two force will cancel, the two force and the mass are the whole time, MG is obviously pulling this thing down and because the spring is being compressed down it pushes the block back up with a force of the spring which equals Kx at equilibrium these forces cancel, that's the definition of equilibrium which means that they will be the same magnitude and I can write Kx=Mg and I hope you remember this Kx=MG, so when I'm doing Part A and I want to know what is Xb? This is how I'm going to start, Kxb equals MG because it's an equilibrium, B is at equilibrium so Xb is just MG/K.mass.gravity I'm going to use 10, k is 800, Xb therefore is 0.125 meters, that's the first answer, and I'm going to put it right here so 0.125 meters is my Xb, let's go to part B and I want to make a little separation here for these questions.

For Part B, I want to know what is the block's maximum speed and where does it happen, Ok? So, the block starts a point C right here I pull this thing all the way down and I release that's going to start going up, it's going to start accelerating up so its velocity is getting faster and faster and faster and the reason why the velocity is getting faster is because at point C the Kx is much bigger than MG, draw a little tiny MG, Right? So, if you pushing the spring down, the spring is really pissed off and it's pushing back up so this thing will accelerate up, the problem is once you cross your equilibrium, right? Equilibrium is right here so at this point you are being accelerated up once you cross your equilibrium the forces are the same your KX is no longer greater It's actually equal and once you go above equilibrium you're KX is basically getting weaker and weaker and weaker so the force that pushes you up keeps getting weaker once you cross.... Keeps getting weaker and it's weaker than the weight force once you cross point B, OK? So, you have an acceleration up but then you go to equilibrium which is 0 and then you actually start accelerating down which means you start slowing down so the fastest you will ever be is at when you cross the equilibrium so I want you to remember that and I'm going to actually write that down here, this is for the other part of the problem but V Max happens at equilibrium, OK? Please remember that on the vertical spring V Max happens at equilibrium so when I ask you for V Max what I'm really asking in this particular case is Vb which is at equilibrium, now I have a spring moving from C to B and if I want to know the velocity of a moving spring on a vertical spring like this we're going to use energy, right? And if it was a horizontal spring it would be the same thing but when you have a moving block spring and you want to find out information you use energy, OK? We're going from C to B so I'm going to write Kc+Uc+work non-conservative=Kb+Ub, it might seem weird and it is unusual that we're going from C to B usually we go A B and C but now we're going back C B and A, right? So, there is no kinetic energy at point C because there's no speed, potential energy there are two types I have Uc gravitational and I have Uc elastic, is there a gravitational potential energy? I want you to try to get these because identifying this is the most important part so is there gravitational potential energy? Hopefully you said no because there's no height, is there elastic potential energy? By the way I wrote UGEL, It's UC El potential energy at point c, there is elastic potential energy because the spring is compressed, right? So, I have this because the spring is compressed, there's no work because once I push down I do work, right? When I push down I do work but then I release and this thing is moving on its own. So, from B to C, I'm not doing anything I did some stuff before but it doesn't matter...It doesn't matter on this question here so there's no work non-conservative there's also no friction so there's no....Both types of non-conservative work don't exist, kinetic energy at point B there is a velocity point B you're accelerating in fact it's the high speed so this exists and again here there could be two types, there's UbG And UbEL Is there gravitational potential energy at point B? Yes, because it is not your lowest point, is there elastic potential at point B? Yes because it's not the original length of the spring even though you're at equilibrium it doesn't mean that you don't have no compression you've compressed down a bit in fact you compressed down 0.125, so now what we have to do is replace all these numbers and we'll be good so over here I have 1/2K (which is 800) and then X squared, the compression at point C I hope you see right here the compression at point C is the entire compression which is 0.425, this is 1/2M(I can plug in the mass, the mass is 10)Vb squared, this is gravitational potential energy which is M (10) G (10), what is the height at point B? If this is at 0 the height at point B is just 0.3, right? hope you see that and IÕm going to put this other guy over running out of space there, 1/2KX squared, what is the compression of the spring at point B? Look at diagram and tell me what the compression of the spring at point B is? So hopefully you said it is 0.125, right? This is the compression so the height is this but the compression is this, the height is relative to the bottom and the compression is relative to the original length, right? So those are distinct numbers, cool. So now if you notice we have a bunch of numbers and Vb is the only variable so the physics is done we just have to solve here using algebra and in the interest of time here you should definitely practice this and make sure you get the answer right but the interest of time I'm going to skip some steps here and tell you that Vb comes out to be....2.68 meters per second and that is your maximum speed. Alright so let's go to part C, Part C is to find a velocity when you're back to your original height, your original right here so we want to know what is the velocity at A? So, for parts C, you want to know what is the velocity at point A? Again moving spring so we're going moving block and spring so we're going to use energy I could use energy from A.....From C to A or from B to A whichever you prefer, C to A is easier because there are less types of energies here, the total energy is conserved but there are less types of energies from C because there is no kinetic energy, B has kinetic energy it has potential energy B's just a huge mess so it's easier to go from C to A, So Kc+Uc+work non-conservative=Ka+Ua, is there kinetic energy at point C? No, what are potential energy at point C? Potential gravitational and potential elastic, is there gravitational point C? No because there is no height, is there elastic? Yes, because there is spring compression, there's no work non-conservative because once you release you're not doing anything and there's no friction, kinetic energy at point A, the object is moving right so it's going to come back and shoot past point A so there is kinetic at point A and there is potential energy at point A, now is that gravitational? Is that elastic? Or is it both? Is there a gravitational potential energy at point A? I hope you say yes because there's a height, is there elastic potential at point A? I hope you say no because the spring is back to its original height, back to its original height means that it is back to being uncompressed, those are the types of energies what is Uc? Uc is 1/k(K=800) X squared, X is the compression of the spring at point C, look at the diagram and see if you can figure out what's the compression at point C? The compression at point C is the addition of these two numbers just like before so 0.425, over here I have half the mass is 10VA, VA is what I'm looking for and here gravitational potential M=10, G=10, H let's figure that out what is the height at point A? Look at the diagram I hope you agree hope you sad that the height at point A 0.424, the height at point A is the same as the compression at point C and that should make sense all the way at the bottom you're fully compressed all the way at the top you have no compression at all, you have all height so here are all height no compression here all compressional no height, they essentially swap, if you look here you have everything you need except in and only one unknown so you can solve this, again in the interest of time I'm going to skip out the algebra here that part is done with the physics so if you do all this you get the VA=2.44 meters per second, now one way that you can tell that this answer is probably correct one way that you can have a little more certainty around this answer is the fact that this number came out to be less than your VMax down here, V max is 268, this is less than your V max which is good, I will put a little good check mark there, if you got a bigger number than something's wrong because the 2.68 was supposed to be your bigger number so that's kind of good, let's go to part D and we'll make it fit right there so what is the maximum height above the original height? So, we're going from A to D and I want to know not the height from the bottom I want to the height from the top of A, the height from the top of A so I want to know what is Hd? And the height at A since it's the lowest point we'll just call it 0, OK? So, it's an energy equation from A to D, right? On the way back and we want to know what is Hd? So Ka+Ua+work non-conservative=Kd+Ud, this is the easiest part of the question there is a kinetic energy at point A because you're passing with a velocity, there is no potential at point A that's because we're saying that's the lowest point between A and D so the height 0 and the spring is not compressed at that point, work non-conservatives is also 0 because we're not doing anything no friction, there is no kinetic energy at point D because it's the highest point so it stops and at point D there is only gravitational potential we have a height so we have gravitational but we have no spring compression the spring's....the block's already off the spring so there is no elastic, OK? So the only thing you have is all speed at the bottom all height at the end so I have Ka=U gravitational D, So this is 1/2MV squared=MGHd, the masses cancel and then Hd=Va squared /2G, and Va is 2.44 squared/2(10), I actually didn't calculate this number in advance.....Just a second.....But it should be..It should be 0.295, I could be off by a little bit you can do this in your calculator again the most important part was getting right before this point here if you got here then it's just a matter of plugging into the calculator, Cool? So that's it we got all the answers, this question is really long you probably wouldn't get something like this in a test even if you have a long answer there's too many parts but you could get two or three of these and what I wanted to do with this example and the previous examples is to cover just about every possibility, every reasonable possibility you can have your professor can all do crazy stuff but this covers the vast majority of it, OK? So hopefully it makes sense let me know if you have any questions.

A spring with spring constant *k* oscillates with a total mass *m*. The shortest time interval between the instant of greatest speed and the instant when it turns around is

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It takes 4.36 J of work to stretch a Hooke’s-law spring 7.54 cm from its unstressed length. How much the extra work is required to stretch it an additional 4.8 cm?

1. 27.5923

2. 13.8816

3. 14.269

4. 3.54616

5. 6.58426

6. 5.18114

7. 15.988

8. 17.1564

9. 18.2072

10. 7.31815

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