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$\bullet$ $\bullet$ A spring of negligible mass has force constant $k=$1600 $\mathrm{N} / \mathrm{m}$ (a) How far must the spring be compressed for3.20 $\mathrm{J}$ of potential energy to be stored in it? (b) You place thespring vertically with one end on the floor. You then drop a$1.20-$ -kg book onto it from a height of 0.80 m above the top ofthe spring. Find the maximum distance the spring will becompressed.

(a) 6.30 $\mathrm{cm}$(b) 11.6 $\mathrm{cm}$

Physics 101 Mechanics

Chapter 7

Work and Energ

Physics Basics

Applying Newton's Laws

Kinetic Energy

Potential Energy

Energy Conservation

Cornell University

University of Washington

Simon Fraser University

University of Winnipeg

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04:16

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okay, Since the potential energy is given in our problem, we know that potential energy is equal to half off key excess squared where cases spring constant and X is the compression which we need to find. Now. If we write the expression in terms of X, we see that X is equal to square, rode off to you over key. And if we use, they're not given numbers. We see that you is three point toe Jules. He is 16 100 meter. Using that we find the compression as zero point 0632 meter or in centimeters. It's 6.32 centimetre. Now, in the second part of the problem, we've drawn two different scenarios. One is when the book is at a height 10.8 meters from the top of the spring, and then in second scenario of in the spring is compressed, played a seance. De um, the book is at H plus de meters below its initial position. So as we can see that we can use energy conservation to solve this problem. So we need to find the compression from there are we need to find d from this set up. If we used the energy conservation. We see that since this's one. So let's write all the energy's associated with 0.1, so we'll have key one plus you want plus W other which is equal to the second conversation which will be Kito plus you too. Okay means the kinetic energy and you means thie potential. Indigent W is some other former a form of energy which we ignore here because all we have in the form of potential energy is the gravitational potential energy and in the form of kinetic energy, we just have having V squared. And ah, we are neglecting the air, drug and other forms of energy so we can safely ignore w other. So we can say that 20 And because of that, we have key. One plus you won is equal to he told us, you two. Now to find out, you tow and find out these parameters. First, let's fix the origin and then we can think about how to solve those now for calculating the potential energy. We can easily see that since there's a height, it will be the the gravitational potential energy, which is mass Times square 30 times the height. How do we fix the height? So it's up to us because the way we choose the origin determined the gravitational potential energy and the way we're choosing it is we're setting the height around here where the book is compressing the spring. So are setting that point as why I called to zero and then vertically upward is vertically upward. Direction is they hide that the book has so basically from here tto desperation. Where the book was initially book was initially is H plus d. So that means the rings gravitational potential energy at that point will be m times two times h plus de. But there will be no kinetic energy because we see that the velocities here over here so we can safely ignore the kind of energy at that point. Similarly, since the book comes to stop at this point, that means in the velocity become zero. So we can safely ignore depended the energy in the second point as well. And for the potential energy, it will be the amount of compression. So since we know that the amount of compression is D, so that's going to be key times D squared now this K and that is different. This case, the spring constant. And this one is the kind of energy Tio let's call it small King so that we can differentiate between this toe. So yeah, the potential at this area will be the spring constant times distance squared since the initial without the compression that spring was here and it compressed. The amount of compressed is thie, so the amount of attention and restored in the spring we have kids world. Similarly, if we take from this sin position tau disposition off the book, then we see that the total height travel is inch plus D. So that means the gravitational potential energy on the book will be M times t times a tch trustee. Compression. Right? So let me write that equation. Don't one more time. So we have m times g times H plus De is equal to half ki d squared and let's write everything on one side and said that equals zero producer. We have half Keeley squared minus M d d. When it's md h zero. From here, we can see that this is a formal quadratic equation because we're solving for this unknown Part D and D has a square term over here, so that's a quadratic form of decoration. And we know that since its aquatic form, we can solve the and get the following expression so they will be won over key times mg. And there's a plus minus sign. Since it's a quadratic. So there will be two solutions. One is the positive +11 is the negative one. Then we have M d squared plus four times half off key. Thanks N G H. Now notice one thing. This expression is always greater than md on. The reason for that is because we have an M d squared over here, so ah, there's a square root and there's a square terms. So anyway, this term is there. Then we have plus some other time, which is no on non negative or which is positive. So that means we have md plus sometime that's added to it. So we can easily say that this whole thing is greater than empty, so that means if we take the negative solution here will get Dia's negative. But we don't want that since we have already said that he's positive, so we'll only conservative positive solution. So we'll take the positive solution. And if we do so and use the numbers, so let me write down their expression one more time. We're just taking the positive solutions. So that will be won over K times nd for us. And he squared plus four times, half off key times and and we're already given all the values. So let's use them. So it's one over 16 100 Newton per meter times one point toe k g times the gravitational expiration 9.8 meters per second squared plus the same thing 1.2 kg Thanks 9.8 meters per second squared in this square on all the dance. Then we have for times one over two, 1,600 newtons per meter times uh, 1.2 kg times 9.8 meters per seconds squared times Uh, the height that is 0.8 meter. Using all them numbers, we see that d is 0.74 meter plus you know point 1087 meter which gives us a total of serial 0.12 meter on 12 centimeters. So that means the amount of compression that the spring hand was 12 centimeters. Thank you

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