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a block is initially at position X equals zero and in contact with an uncompressed spring of negligible mass the block is pushed back along a frictionless surface from position x equals 0 to x equals negative D as shown above compressing the spring by an amount Delta x equals d so the block starts here and it's just in contact with the spring so it's initially the spring is uncompressed and it's just touching the block and then we start to compress the spring by pushing the block to the left and we compress it by an amount D they tell us that right there Delta X is equal to D so we compress we move this block back over to the left by D and that compresses the spring by D the block is then released at x equals 0 the block enters a rough part of the track and eventually comes to rest in it position x equals 3 d so when we compress the spring we're actually doing work to compress the spring and so that work that energy from that or the work we're doing get stored as potential energy in the spring block system and then when we let go that potential energy is going to be converted to kinetic energy and that block is going to be accelerated all the way until we get back to x equals 0 then the spring is back to uncompressed so it's not going to keep pushing on the block after that point and then the block is going to have this kinetic energy and if there was no friction in this gray part here would just keep on going forever and if there's no air resistance and we're assuming no air resistance for this for this problem but since there is friction it's just going to decelerate it at a constant rate you're going to have a constant force of friction being applied to this block so let's see they say they tell us and it's going to come to rest at x equals 3 d the coefficient of kinetic friction between the block and the rough track is mu all right on the axes below sketch and label graphs of the following two quantities as a function of the position of the block between x equals negative d and x equals 3 d you do not need to calculate values for the vertical axis but the same vertical scale should be used for both quantities so they have the kinetic energy of the block and the potential energy of the block spring system so let's first focus on the potential energy you because when we start the first part of this when we're compressing the spring that's when we're starting to put potential energy into this spring block system and so you have to think about what is the potential energy of a compressed spring well the potential energy the potential energy is equal to 1/2 times the spring constant times how much you compress the string this the spring squared so if we want to say Delta X is how much you compress the spring that squared now if what I just wrote is completely unfamiliar to you I encourage you to watch the videos on Khan Academy on the potential energy of a compressed spring or the work necessary to compress a spring because a work necessary to compress the spring that's going to be the potential energy that you're essentially putting in to that system and so for this as we compress the spring to D you are you're going to end up with a potential energy of one-half times the spring constant times our change in X is d our change in X is d times D times d squared so let's plot that on this right over here so right what whoops right when we are at x equals zero there's no potential energy in our system but that we start to compress it and when we get to x equals D we're going to have a potential energy of one-half times the spring constant times d squared so let's just say this right over here let's say that over there actually let me do a see that one is actually I'll do it over here so it will be useful for me later on so let's say that this right over here is 1/2 times our spring constant times d squared so this is what our potential energy is going to be like once we've compressed the spring by D and it's not going to be a linear relationship remember the potential energy potential energy is equal to 1/2 times the spring constant times the spring constant times how much you've compressed the spring squared so the potential energy increases is the square of how much we compress the spring so when we've compressed bring half as much you're going to have 1/4 the potential energy so it's going to look like this it's going to be you could view it as the left side of a parabola so it's going to going to look something something like this so that's the potential energy now when you're in this point when the the thing is fully compressed and then you let go what happens well that potential energy is turned into kinetic energy so as the spring as the spring accelerates the block you're going to go down this potential energy curve as you go to the right but then it gets converted to kinetic energy so the potential energy plus the kinetic energy needs to be constant at least over this period from x equals negative D to x equals zero so the kinetic energy starts off at zero it's stationary but then it starts the block starts getting accelerated it starts getting accelerated and the sum the sum of these two things needs to be equal to 1/2 times our spring constant times d squared and so you could see if you if you were to add these two curves at any position you are going their sum is going to sum up to this value and so right when you get back to x equals 0 all of that potential energy has been converted into kinetic energy and then that kinetic energy it we would stay at that high kinetic energy if there was no friction or no air resistance but we know that the block comes to a rest at X is equal to 3 D so all the kinetic energy is gone at that point and you might say well what's that getting converted into well it's getting in converted into into heat due to the friction so that's where you know energy cannot be cannot be created out of thin air or lost into thin air it's converted from one form to another and so the question is what type of a curve is this do we just connect these with a line or is it some type of a curve and the key realization is is that you have a constant force of friction the entire time that the block is being slowed down the coefficient of friction doesn't change so the force of friction or the mass of the block isn't changing so the force of friction is going to be the same and it's acting against the motion of the block so you can you can view the friction as essentially doing this negative work and so it's sapping the energy if you think about it relative to distance in a given amount of distance its sapping away the same amount of energy it's doing that same amount of negative work and so this is going to decrease at a linear rate and so let me draw that so it's going to be a linear decrease just like that and the key thing to remind yourself is is this is a plot of energy versus position not velocity versus position or velocity versus time or energy versus not time this is energy versus position and that's what gives us this linear relationship right over here so we have the kinetic energy K of the block that's what I did in magenta so this is the kinetic energy kinetic kinetic energy and in blue just to make sure I label it right this is the potential energy potential potential energy

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