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Directions: Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part.

(Figure is not available to copy)

An experiment is designed to calculate the spring constant $k$ of a vertical spring for a jumping toy. The toy is compressed a distance of $x$ from its natural length of $L_{0}$ , as shown on the left in the diagram, and then released. When the toy is released, the top of the toy reaches a height of $h$ in comparison to its previous height and the spring reaches its maximum extension. The experiment is repeated multiple times and replaced with different masses $m$ attached to the spring. The spring itself has negligible mass.(a) Derive an expression for the height $h$ in terms of $m, x, k,$ and any other constants provided in the formula sheet.(b) To standardize the experiment, the compressed distance x is set to 0.020 m. The following table shows the data for different values of m.

(Table is not available to copy)

(i) What quantities should be graphed so that the slope of a best-fit straight line through the data points can help us calculate the spring constant k?(ii) Fill in the blank column in the table above with calculated values. Also include a header with units.(c) On the axes below, plot the data and draw the best-fit straight line. Label the axes and indicate scale.

(d) Using your best-fit line, calculate the numerical value of the spring constant.(e) Describe an experimental procedure that determines the height h in the experiment, given that the toy is only momentarily at that maximum height. You may include a labeled diagram of your setup to help in your description.

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Chapter 12

Practice Test 2

Section 2

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many examples of conservation of mechanical energy involve spring potential energy getting converted into other forms. So typical examples might be clowns getting shot out of spring cannons during a circus event, um Or toy guns that can shoot marbles upwards. Um And in these cases you're converting spring potential energy into gravitational potential energy or and or kinetic energy. But let's take a concrete example, let's suppose you have one of those little spring toys, they usually look like chickens or rabbits, but we'll pretend it it's just um a mass. And will also say that you can load different masses onto the toy. The spring starts off relaxed with a relaxed length L. Zero, which will change as you load the toy down. But we'll have uh some way of measuring that, we'll say. So you squeeze the toy down, press it down a little bit. Let's say you always maintain the same distance through which you're going to compress that spring. We'll call that distance X. And what you're doing with your experiment is you're trying to determine the spring constant K. Of that spring. And it's a little bit too hard to do sort of the vertical hanging experiment. So what you're going to do instead is watch the little toy leap up into the air and it will leap up to a certain height after the spring completely relaxes and leaves the surface. Um And that height will be measured relative to where it started with compression by delta X. Or by X. We'll call it. And the question is, how can you experimentally determine the spring constant from these experiments? Well, this is conservation of mechanical energy. And the form that you're starting with initially is the um spring potential stored in the spring. And that's an amount one half K. X. Squared. And the final state of energy is gravitational potential. MGH Provided you measure that h um upstairs from the low point of compression. Okay so that's just equal to MGH. Ah um Now you could just take one data point uh and solve it for the spring constant knowing all the other parameters. But of course it's better to do an experiment. So the experiment is going to have us load different masses onto the toy. Um And for each mass you're going to somehow determine how high the toy grows with that mass loaded on there. And a way to do that is you would want to um put a ruler a meter stick, ruler depends how high it goes behind the toy and perhaps record with your phone so that you can go look at the videos later and catch at what marking the mass rose. And you can play through those videos slowly so you can see what that marking is more clearly. Um But in any event let's suppose you do that experiment and you gathered some data. Um So I'll share some data. Um But what you would like to do is to make a plot of some sort in order to make a straight line and determine the spring constant from the slope of that line. So let's see in your experiment will say that you're controlled variable is the mass that you're loading on. So that's what's called the independent variable. Um And H. Is the dependent variable. It's the result of the experiment. Um So you basically want to get H. As a function of em. And so let's divide both sides of that equation by M. M. I. G. And you get Kay X squared over two MG. Is equal to age. Now that won't make a very good straight line. Um Remember you're holding the x. constant and of course G is just 9.8. Um But because amazon the denominator that's an inverse relationship. So the plot that you actually want to make to get a straight line for your experiment would be H versus one over M. Uh And then we expect as the mass gets larger the height will get smaller and vice versa and we would expect a nice straight line relationship. Um So I'm going to share a table of data and I've made the corresponding plot of the height versus one over the load mass. And let's see uh what we get from that, I'll share that. Right? And I've grafted using Excel, I do like Excel. Yeah. Uh And if you don't have Excel, one of the things you can do is use google sheets, which is freeware if you have uh Uh huh. A google account um you should be able to use google sheets. So indeed when I plot the height in meters vs. One over the mass and inverse kilograms. Indeed, there's a straight line um the data table is shown above. Um And what I can do with Excel, which you can also do my hand is I can fit what's called the best fit straight line, which tries to get as close to as much of the data points as it can. So some of the data will fall a little bit below the line and some a little bit above the line. Um I actually forced my graph to go through 00 as well. Um So that is one of the other points on there. And I find that my slope is 0.0101 with units of meters um per Uh one over kg, so that's a little bit cumbersome. Okay, but the slope From my graph is 0.0101 beaters times kilograms, we'll call it that's meters times kilograms. Um And now we want to solve this for the spring constant, but we can see that in the formula above the quantities that make up the slope, R k, X squared and to G. And that has to equal 0.0101 um giving us the K and S. I. Units. Um So we simply have to cross multiply and know what the initial compression was. So let's say we were starting with the compression of .02 m and we tried to control that, that should not have been too hard to do. Um And so we can solve for that unknown spring constant from slope um and X squared and twice times 9.8. And because everything is in units of S. I meters, kilograms and seconds, the units for k should be newtons per meter. Um and putting all that together, let's see what we get. Uh huh. It looks like it's going to be surprisingly large, surprisingly stiff um spring constant And I'm getting about close to 500 newtons per meter. Um I'll write it down with all its decimal Figures for 95 Newtons Per meter. Probably not that precise, but um very close to 500 newtons per meter, so it's a pretty stiff spring. And if you've ever tried to punch these little toys down, that may not be too surprising.

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