A spring of negligible mass has force constant k = 1600 N/m. (b) ...

Question

A spring of negligible mass has force constant k = 1600 N/m. (b) You place the spring vertically with one end on the floor. You then drop a 1.20-kg book onto it from a height of 0.800 m above the top of the spring. Find the maximum distance the spring will be compressed.

Accepted Answer

Okay, so we have a spring and we're gonna rest it vertically and then we're gonna have a block above that spring. We're gonna drop the block and we want to know how much that spring is going to compress when the block falls. So the first thing we want to do, the problem like this is to draw out what we're given. Okay, So initially we have a spring here with the spring constant given in the question by K equals 1000 newton per meter. We also have this block up here With a mass of 0.5 kg and it's at rest. Okay, So the initial velocity of that block is gonna be zero, it's not moving, Here's the mass and we're told that it is 1.5 m above the spring from the top of the spring. So this distance here between the block and the top of the spring will call it h And it's gonna be 1.5 m. All right, after the block falls, the spring is going to get compressed. Block is going to be on top of the spring, the mass of the block, it's gonna stay the same. Okay, the velocity here, we're looking for when this spring is max the maximum compression on that spring. So we get maximum compression, the block momentarily will be stopped. So the final velocity will also be zero. Okay, And what we wanna know again that maximum compression. So we want to know what's the distance from where the spring started. Way up here, to where the spring finished this distance here and we'll call it D. Okay so that's the maximum compression that we're looking for. Alright. We're just gonna label one more thing on this graph. We're gonna call this minimum point of the spring. Well call that zero. Okay. And then we can base everything off of that. All right. So in this problem we can note that we don't have any concern non conservative forces. Okay. So there's no friction or any other forces. Any other non conservative forces like that. So we can look at the conservation of mechanical energy and our mechanical energy consists of what? Well it consists of the kinetic energy and it also consists of potential energy. So if mechanical energy is conserved, the sum of the kinetic and potential energies before will be equal to the sum of the kinetic and potential energies after. Alright, what are those potential energies in this case? We have to we have the potential energy of the spring. Should be a plus. Not an equals. Okay. And we also have the gravitational potential energy and the same thing on the right hand side. The spring potential. The gravitational potential energies. Alright, great. So let's work out these values. Okay well K. We know that K. Is one half M. V. Squared. Okay that's the potential or the kinetic energy. Sorry. The spring energy is going to be given by one half K. X. Squared. And the gravitational potential energy is given by M. G. H. Okay. Same thing on the left hand side and be not or B F squared. Okay, next word. M G H. Alright, great. Now this looks pretty messy right now but we're going to simplify it quickly here. So one thing to remember is that our velocities are zero. The initial and final velocities are zero. That means these terms both kinetic energy terms will go to zero. We don't have to worry about those. Another thing to note is that our spring initially is at rest. We're told in the question it's not been compressed so it's not storing any potential energy. So here are spring potential will be zero. Okay. And the last thing we want to realize is that in the final case when that block is that the compressor spring maximally that's our height zero. Okay, So it will be at height zero. Which means that it will have no gravitational potential energy. Okay, so we're left with just two terms on the left, we have the gravitational potential energy and on the right we have the kinetic energy or the sorry, the spring potential energy. Okay, so all of that gravitational potential energy that's stored in this block is going to be converted into potential energy stored in that spring. All right, so let's go ahead with the values we know we know the mass. We know the acceleration due to gravity is 9.8m/s and we know that initial height. Now the initial height is not relative to the top of the spring like we were given but actually relative to what we've chosen to be zero. So the initial height of the block is not 1.5 but is actually 1. plus the displacement. D. Okay. And on the right hand side were given our K value and the displacement of that spring is the like we've chosen. Okay, so let's go ahead and simplify. So we get 4.9. 1.5 plus d. Is equal to 500 d. squad. Okay? And just expanding that 7.35, That's 4.90 Equal to de squirt. Alright, we're just gonna move over here to continue our work. And if we move everything to one side, we are left with this quadratic equation that we need to solve for D. How are we gonna do that while we use the quadratic formula? So D. Is going to be negative B plus or minus square root B squared minus four. A. C. All over two. A. Alright. 500 is going to be all right -1.9 is going to be our B and -7.35 is gonna be safe Putting that in whoops, positive. 4.9 plus or minus the square root 4.9 Squared -4. Mhm. See Over two times 500. Who? And if we simplify that out, I'm gonna get 4.9 plus or minus the square root of this number here. We're gonna be divided by 1000. Okay. We go ahead and do that calculation. We're going to get to solutions. We're going to get 0.126. Or we are going to get negative 0.116. Okay. We need to figure out which one makes more sense. So again, we're talking about a spring, we're talking about a maximum compression and we're looking at it as a distance so it doesn't make sense to have a negative. So we're not going to look at that solution and instead this is going to be our solution. Okay. So it's gonna be 0.1 to six m D. That is the maximum compression of that spring when we drop the mass.

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