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Data Analvsisi Sping System Some data has been taken on particular spring: The force required to stretch this spring in various distances is given inthetablebelow: ...

Question

Data Analvsisi Sping System Some data has been taken on particular spring: The force required to stretch this spring in various distances is given inthetablebelow: Displacement (m) Force (N) 12.0 24.0What is the spring constant of this spring?Fill in the missing values of the table above:Predict how much force would be needed to stretch the spring 1.24 m_Sketch plot of Force (y-axis) vs. Displacement (x-axis) using the data table above_Calculate the slope (rise run) of the graph: Include correc

Data Analvsisi Sping System Some data has been taken on particular spring: The force required to stretch this spring in various distances is given inthetablebelow: Displacement (m) Force (N) 12.0 24.0 What is the spring constant of this spring? Fill in the missing values of the table above: Predict how much force would be needed to stretch the spring 1.24 m_ Sketch plot of Force (y-axis) vs. Displacement (x-axis) using the data table above_ Calculate the slope (rise run) of the graph: Include correct units.



Answers

DATA A physics student measures the energy stored in a spring as a function of the distance it is stretched beyond its undistorted length. Her data are given in the table. $$ \begin{array}{cc} x(\mathrm{~cm}) & \text { Energy (J) } \\ \hline 2.6 & 0.34 \\ 6.3 & 2.00 \\ 7.5 & 2.81 \\ 8.2 & 3.36 \end{array} $$ Draw a linearized graph of the data by plotting the spring's energy as a function of the square of the distance it is stretched. Using a linear "best fit" to the data, determine the force constant of the spring.

To solve this problem. We're going to set up a graph that has a Y axis of Newtons of force applied to spring and in X axis of meters that the spring is stretched giving and force applied to it. When we look at a graph like this, we recognize that the slope of this line is going to be in units of Newtons per meter, which is our units for que a spring constant. So, in order to solve for the spring constant, all we need to do is sell for the slope of a graph and we've been given points on that graph. So we can do that, knowing that K is going to be a change in Newton's over change in meters. We're going to use two points. We have five Newtons minus one Newton over point 285 meters minus 0.145 meters have converted those from centimeters and commuters and wins. We plug this in, we find that our spring has a spring constant of 29 newtons per meter. And now you're going Teoh Ah, use that spring constant to that slope to help us set up in creation for this graph. So we know that why equals MX plus B, that is equation for linear graph. In this case, Y is equal to the force of the spring. In new tens, he's slope and we have just solved to be 29 Newtons per meter. We are. We have, uh, exes distance in that the spring is stretched in meters and then be is our value Holmes of our y intercept. So we're gonna plug in, um, an X and A Y coordinate to solve for the y intercept. So we're going to use five Newtons in 50.285 meters with our coordinate points on a graph. And so we have five equals 29 times 0.285 plus B in order to solve for B. And we find that B is equal to negatives three 0.1 for so now, even equation for our graph, it is force of spring equal to 29 x. That's displacement of the spring minus 3.14 And we can use this to solve for our value of eggs springs displacement. When we have a force pointed spring of zero, that's gonna be the spring at rest Ciro Force applied, Teoh equals 29 x minus 3.14 We can solve this equation for X, the distance that the spring is displaced and we find that it is 0.1 099 meters, which is equal to 11 centimeters. So our spring has an equilibrium position length of 11 cents.

This triple shows the force forces exchanging stranger down Begin See there at the initial residual value The extension aciego when RIA played a tune. Urgent Poor sex extension is about 15 millimeters and so on and we are having the maximum value value when we apply 22 nutrient pools. We are having the exchange in off 1 90 degree, which is over but one no more to the second part. They're chills the best straight line for Did you endanger? So here we can see that again. We're having DeGraff or force verses The extension in the string which is in the form of the land. The unity Zeman imager. We can see that by joining all the values we over here, we are able to get de exit. Strickling. This gives the exactly data No moving photo in the tart apart. Buy less square. Putting it step is almost are It's a slope is almost we can find from the formula. Why do minus craven up one extra minus X one. It's nothing but their difference off my under wordy Galaxies and the difference off eggs on the horizontal axis. Excuse the answer. A zero point 116 Nugent. Bush major. No Drew draws a straight line. We use all the points listed and also the origin. If the calls off this spring touched the each other a band or non linearity course, sure, at the bottom and off the graph. If the spring will stretch too far or non literally. Najee de Corps. Sure at the top. And but there is no visible evidence for the blind Intergraph near either, and no, in the immigration F is a question. G eggs The spring Const Rinckey is the slope off the F versus extra. And so here we are. Having this value is G randy No, the force begin calculate as from the formula, their physical to G eggs. Is it good job? 116 Nugent, But meter divided by Zito point 105 major. So we will have 12 point to near drink

So we're trying to draw a graph over here. We're trying to start from zero. Go up to 15. So this is divided into three parts five, 10 and 15. So we have got 2.557 point five, 10 12.5 and 15. This is X in meter. Now the Y axis. He's force in Newton. And this goes up as all the way up to at 15. This is 0.71 at half way. This is a little boring. 35 Okay, that will do. So. 7.5 over here, at which I have a 0.35 notice that disease doubles and a little more so that's the other. So all the other points will be on the straight line over here where this goes 2.71 These goes 2.35 and this Just make the points. Read the other point of the here. Here, here, here is the formal point. I got one here and one there. So now we're just going to use the 00 point and and this point to find the force constant so forth constant will come out to be F over X. So this is just use the halfway point at which forces 0.35 new done. Delighted by XY 7.5 meter. So that will come out to be 0.35 divided by 7.5 zero porn zero for 66 Mutant Paar meter Answer for part A Actually joining the graph waas part A On this is part of the using the find the elastic potential energy when the spring is stretchered by half a meter So here is 0.5 meter point and this point is given as what? Half a meter? 10? Okay, the numbers got a little off, so I'm going to redo the numbers again on we don't need all the numbers for this. The highest display Sprint is airborne 71 on the highest point. He's 15 so this number comes out to be 15 new done divided by 0.71 meter. So this would be 21 point 13 new don't per meter. Now for part C, they're asking potential energy when X equals 0.5 meter for that. Here to find worries. Sit a 0.5. It's right here, so we are dealing with this point over here. Andi That number. He's Ah 10 Newton, 10 Newton. So now the potential energy is even by the area of these cars. Idiot. Off the the hash region, which is half times base, 0.5 meter squared times the base halftime based times the height, he's 10 Newton. So that is five dimes. Half is 2.5 June. That is a total energy stored until half a meter stretch.

For this problem, we will use our cell to take the data provided, create a graph, calculate the slope of the line of best fit. And use it to inverse and information about the physical system to begin, I've copied in the data and then I will provide a chart. So I will use a scatter plot. Mhm. And we have, let's see, we have force here on the Y are on the X axis and length of extension on the y axis. Okay, This means that if we take a slope will be in your sense of why over X or in other words millimeters per noon. Okay, So needs to make that clear by saying force. That's right. Uh distance. It's a displacement was just first. We're curious what's happening around the range Of 105 mm. 105 mm around here somewhere. So an excel, we just have to like click add trendline and just like that we get a front line and we can even see the equation here. So in this equation we know the slope to be 815. Units here would be millimeters come in. We invert that we can get the force constant. 1/8 21 5, sorry, that would be the spring constant. Uh this would be in units of newtons per millimeter. Yeah. Now, if we get it in news perimeter, this would be the typical sc value for spring constant. So we just take this and multiply here By 1000 because there are 1000. Yeah, so this is the spring constant K. At the heart of the problem. Mm. Uh, Part B says, okay, so as Part B, part C is if the spring is extended to 105 millimeters, what force does it exert on the suspended weight? So we need to use this plot to interpret late the force At 105 mm. Since I've put the distance or displacement here on the y axis is going to be a little bit more challenging. But it's still possible. All we have to do is say 105 Equals 815 x Plus -10556. This is just the line of best fit And we're applying it to 105 mm. We can solve this for X By saying 105 plus 10556. And then we just divide that by the slope. Yeah, A value of 13.01 newtons notice that's a reasonable value because it falls somewhere between 12 and 14 here in our table. So this is the final answer along with the spring constant. And the table


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