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# Spring potential energy example (mistake in math)

## Video transcript

welcome back so let's do a potential energy with problem with a compressed spring so let's let's let's make this an interesting problem so let's say I have a loop-de-loop a loop-de-loop made out of ice and I made out of ice so that we don't have friction so let me draw my loop-de-loop loop-d-loop there's the loop there's the D loop alright and let's say this loop de loop has a radius of 1 meter let's say this is this right here is 1 meter and so of course the loop de loop is 2 meters I 2 meters high and let's say I have a a spring here it's a compressed spring let's say this is the wall this is my spring it's compress so it's all tight like that and let's say it's spring constant K is I don't know 10 and attached to that compressed spring so I have a block of ice can I need ice on ice so I have no friction this is my block of ice shining and let's say the block of ice is I don't know 4 kilograms 4 kilograms and we also know that we are on earth and that's that's important because this problem might have been different if we were on another planet and my question to you is how much do we have to compress the spring so you know let's say that the springs natural state was was here right if there if we didn't push on it and now it's here so what is this distance how much do I have to compress this spring in order for when I let go of the spring the block goes with enough speed and enough energy that it's able to complete the loop-d-loop and and and reach safely to the to the other end so how do we do this problem well well in order to any loop-de-loop problem the hard part is is completing the heart the high point of the loopty-loop right the hard part is making sure you have enough velocity at this point so that you don't fall down your velocity has to off set the downward acceleration in which case and here is going to be the centripetal acceleration right so that's one thing to think about and you might say wow this is complicated Atmos bring here it's going to accelerate the block and then the block is going to get here then it's going to decelerate decelerate this is probably where it's going to be at its slowest then it's going to accelerate back here it's a super complicated problem and in physics whenever you have the super complicated problem it's probably because you are approaching it in a super complicated way but there might be a simple way to do it and that's using energy potential and kinetic energy and what we learned in when we learn about potential and kinetic energy is that the total energy in the system doesn't change it just gets converted from one form to another so it goes from potential energy to kinetic energy or you know or to heat and we assume that there's no heat because there's no friction so let's do this problem so what we want to know is how much do I have to compress the spring so what I'm essentially saying is how much potential energy do I have to start off with when with this compressed spring in order to make it up here so let's what's the potential energy let's say I compress the spring X meters so I compress the spring X meters and in the last video how much potential energy would I then have well I we learned that the potential energy of a compressed British spring and I'll call this the initial potential energy the initial potential energy with an eye is equal to 1/2 K x squared and we know what K is I told you that the spring constant for this spring is 10 so my initial potential energy is going to be 1/2 times 10 times x squared so 10 x squared so what are all of the energy components here well obviously at this point the block is going to have to be moving in order to not fall down so it's going to have some velocity V is going tangential to the loop-de-loop and it also is going to have some potential energy still and where is that potential energy coming from what's going to come it because it's it's up in the air it's it's above the surface of the loop-de-loop so it's going to have some gravitational potential energy right so at this point we're going to have some kinetic energy we'll call that well just call that kinetic energy final because this is why we care about although maybe you know here might be the kinetic energy final but I'll just define this as kinetic energy final and then let's plus the potential energy final and that of course has to add up to 10 x squared and this of course now this was kind of called the spring potential energy and now this is gravitational potential energy so what's the energy at this point well what's what's the kinetic energy kinetic energy final kinetic energy final is going to have to be 1/2 times the mass times the velocity squared right and then what's the potential energy at this point its gravitational potential energy so it's the mass times gravity times this height right so potential and so I'll write that here potential energy final is going to be mass times gravity times the height which also stands for Mass General Hospital anyway why you can tell my wife's a doctor so I know my brain just I don't know anyway so let's figure out the kinetic energy at this point so what does the velocity have to be well we have to figure out what the centripetal acceleration is well and then and then given that we can figure out the velocity because we know that the centripetal acceleration and I'll change colors for variety centripetal acceleration it has to be the velocity squared over the radius right or we could say and and what what is this interpreted acceleration at this point was just the acceleration of gravity 9.8 meters per second squared so 9.8 meters per second squared is equal to V squared over R and what's the radius of this of this loop-de-loop well it's 1 so V squared over R is just going to be equal to V squared so if V squared equals 9.8 we can take the square root or we could just substitute the 9.8 straight into this equation right so the kinetic energy final kinetic energy final is going to be equal to 1/2 times the mass times four times V squared times 9.8 and that equals that equals let's just say let's just use G for 9.8 because I think that might keep it interesting so this is just G right so it's 2 times G so the kinetic energy final is equal to 2 G and you know G is normally kilogram meters per second squared but now where its energy right so it's going to be in joules but it's 2 G all right and what is the potential energy at this point well it's the mass which is 4 times G times the height which is 2 so it's equal to 8 G right so what's the total energy at this point the kinetic energy is 2 G the potential energy is 8 G so the total energy at this point is 10 G 10 G total energy 10 G so the total energy at this point is 10 G and we didn't lose any energy to friction and heat and all of that so then the total energy at this point has also got to equal 10 G and at this point we have no kinetic energy because this block hasn't started moving yet so all of the energy is of potential energy so this also has to equal 10 G and this G I keep saying is just 9.8 I just wanted to do that just so you see that it's a multiple of 9 point a just for you to think about so what do we have here all these numbers worked out well so let's divide both sides by 10 you get x squared is equal to G which is 9.8 so the X is going to be equal to the square root of G which is going to be equal to what let's see if I take 9.8 and take the square root of it it's like 3 point 1 3 so X is 3 point 1 3 so in order so we just did a fairly what see me what seemed to be difficult problem but it wasn't so bad we just said the whole day with the energy in the beginning has to be the energy at any point in this assuming that none of the energy is lost to heat and so we just figured out that if we can press compress the spring with the spring constant of 10 if we compress it 3.3 me three point one three meters we will have created enough potential energy and in this case the potential energy is 10 times 9.8 so roughly 98 joules 98 joules of potential energy to carry this object all the way with enough velocity at the top of the loop-de-loop to complete it and then come back down safely and so if we want to think about it what's the kinetic energy at this point well we figured out it was 2 times G so it's like nineteen point six joules or is that you know 9.19 point six right and then at this point it is it is 98 joules all right did I do that right well anyway well I'm running out of time so I hope I did do the last part right but I'll see in the next video