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In an experiment, students are to calculate the spring constant k of a vertical spring in a small jumping toy that initially rests on a table. When the spring in the toy is compressed a distance x from its uncompressed length L0 compression. The students repeat the experiment several times, measuring h with objects of various masses and the toy is released, the top of the toy rises to a maximum height h above the point of maximum taped to the top of the toy so that the combined mass of the toy and added objects is m. The bottom of the toy and the spring each have negligible mass compared to the top of the toy and the objects taped to it.

 

(a) Derive an expression for the height h in terms of m, x, k, and fundamental constants. With the spring compressed a distance x = 0.020 m in each trial, the students obtained the following data for different values of m.

(b) 

i. What quantities should be graphed so that the slope of a best-fit straight line through the data points can be used to calculate the spring constant k? 

ii. Fill in one or both of the blank columns in the table with calculated values of your quantities, including units. 

(c) On the axes below, plot your data and draw a best-fit straight line. Label the axes and indicate the scale.

(d) Using your best-fit line, calculate the numerical value of the spring constant. 

(e) Describe a procedure for measuring the height h in the experiment, given that the toy is only momentarily at that maximum height.

1 Answer

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Best answer

(a) Apply energy conservation. All of the spring potential becomes gravitational potential 

Usp = U

½ k ∆x2 = mgh 

½ kx2 = mgh 

h = kx2/2mg

(b) You need to make a graph that is of the form y = m x, with the slope having “k” as part of it and the y and x values changing with each other. Other constants can also be included in the slope as well to make the y and x variables simpler. h is dependent on the different masses used so we will make h our y value and use m as part of our x value. Rearrange the given equation so that is it of the form y = mx with h being y and mass related to x.

We get

y = mx

h(kx2/2g)1/m

so we use h as y and the value 1/m as x and graph it.

(note: we lumped all the things that do not change together as the constant slope term. Once we get a value for the slope, we can set it equal to this term and solve for k)

1/m m (kg) h (m)
50 0.020 0.49
33.33 0.030 0.34
25 0.040 0.28 
20 0.050 0.19
16.67 0.060 0.18
X values Y values

Solving gives us k = 490N/m

(d) The slope of the best fit line is 0.01 We set this slope equal to the slope term in our equation, plug in the other known values and then solve it for k

0.01 = (kx2/2g)

0.01 = (k(0.02)2/2(9.8))

(e) - Use a stopwatch, or better, a precise laser time measurement system (such as a photogate), to determine the time it takes the toy to leave the ground and raise to the max height (same as time it takes to fall back down as well). Since its in free fall, use the down trip with vi=0 and apply d = ½gt2 to find the height. 

- Or, videotape it up against a metric scale using a high speed camera and slow motion to find the max h.

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