Spring Force
Part 1: Determining a force function for a spring.
Recall that the force exerted by a stretched spring varies according to the displacement of the stretch from the spring’s equilibrium position. This means that the more one stretches or compresses a spring, the more it pushes back. This is true in all cases until the elastic limit of the spring is exceeded. Sometimes the force varies as a direct linear relationship with the length of the stretch (or compression). In more unusual cases the force may vary according to some other common mathematical relationship (i.e., x2 or x).
In this experiment you will determine a force-distance relationship for a spring provided by your instructor.
1)Which quantities should be graphed (and on which axes) in order to yield a function that demonstrates the relationship between the force applied by a spring when it is stretched?
2)Design an experiment to test the force-distance relationship for your spring. Be sure to include a list of appropriate materials.
3)Is the relaxed/equilibrium length of the spring constant? How will this affect your data?
4)When constructing your force function, F(x), where F represents the restorative force from the spring and x represents the length of the spring stretch, should the equilibrium length of the spring be included in the data values or should it be subtracted from the measured length of the spring?
Note: Exercise care when measuring the stretch length of the spring. Little errors (i.e., millimeters) can make a significant difference later!
5)Implement your procedure and record your data. You must have at least 5 distinct data units (i.e., force-stretch distance pairs). Seven or eight data pairs are preferable. The more data you collect, the easier and more accurate part II of this experiment will be.
6)Graph your data (on graph paper!)! Describe the shape of the function.
7)What does the rate of change of this function represent? Use a dimensional analysis to answer this question.
8)What is the name of this quantity in physics?
9)Use a valid statistical method that employs all of your data simultaneously to compute this rate of change. (i.e., do not do 5 separate calculations!) Record your result on your graph.
10) What is the value of the y-intercept of your force function? What does this value represent?
11)Ideally what value should your y-intercept be?
12)How can you use your function to predict the amount of force the spring will exert when it is stretched a particular distance? Conversely, can you determine the amount of force needed to stretch the spring exactly 2.25 m?
13)Save your data and analysis for part II.
Use the data below (not your collected data!) to answer the following questions:
A spring is compressed to various distances, x, as shown below and the restorative force of the spring, F, is recorded.
x (m) / F (N)0.00 / 0
0.05 / 4.1
0.10 / 16
0.15 / 40
0.20 / 66
0.25 / 103
14)Determine a force-distance function for this data.
15)Is this a linear spring? Why or why not?
16)The general form of the equation above can be written as
Where F represents the spring force and x represents the displacement from equilibrium. A determines the magnitude of the rate of change of the function (much like the spring constant k for a linear spring).
What quantities should be graphed in order to yield a straight line whose slope could be used to calculate a numerical value of A?
17)Calculate values for any of the quantities identified in number 15 that are not given in the data and record them in the table above. Be sure to label the top of the column, including units.
18)On the axes below, plot the quantities that you indicated in number 15. Be sure to appropriately label all axes.
19)Using your graph, calculate A.
20)How does your new value of A compare with your original coefficient from the force function equation in number 13)?
Spring Force
Part 2: Determining the work done by a spring.
As you explored in part 1, springs create a non-constant force that depends on the stretch or compression distance of the spring. We determined the spring constant k for your linear spring at the beginning of the lab, and analyzed data from a non-linear spring later in the period. In this part of the lab we will examine the work or energy exchanged by the spring in this process. Recall that since the spring force is not constant, the amount of work done in each equal interval of stretch (or compression) will be different. Therefore the equation W= Fx will yield an inaccurate average of the energy exchanged in the process of stretching a spring.
Consider a scenario where a large mass falls onto a standard, linear spring, compressing the spring as it decelerates. Looking at the force equation from yesterday, F=kx, we can clearly see that the force of the spring will vary with the distance of stretch/compression. This means that if we know the force for each stretched position we can calculate work as a finite summation. For the scenario described above, consider:
where F0 < F1 < F2 < F3 < F4 < F5, according to Hooke’s Law. This means that since the spring is compressed farthest at the sixth X, this interval produces the largest force.
We can approximate the work done by the spring over this entire compression as
That is, the work done for each little distance of compression added together yields the total work done by the spring.
1)On your own paper, use your data from the linear spring in part I to approximate the work done in each discrete interval. Remember to use the difference in both the force values and distance values for each data set. See your instructor if you have questions.
2)Add the approximate work values from each interval to find the total work done in stretching the spring.
The method for calculating the work done by the non-constant spring force used above provides only an approximation. In calculus, this type of method is called a Riemann sum. To find the work done exactly, we must find the work done by the spring over each infinitesimally small x and add them all. That is, as x dx, these “chunks” of space under the curve become tiny slivers. The area of each sliver represents the work done in that interval.
Consider your graph from part I. To find work, one must multiply force times displacement. If one multiplies the quantities of force and displacementon the graph, one is really multiplying the values on the vertical and horizontal axes. Geometrically, this means that you are finding the area between your function and the horizontal axis of the graph. In calculus this operation is called an integral or an anti-derivative. The Riemann sum you just performed is an approximation of this operation.
3)Using your graph and function for your linear spring force take the numeric integral of this function from the least stretched value (i.e., zero displacement) to the most stretched value as your upper limit. Follow these steps on your TI calculator to perform this operation.
- Input your force and displacement data into your lists and graph as described above.
- Use the linear regression feature to find an equation of best fit for your data and plug this function into Y1. If the best fit function does not have a y intercept of zero, you will have to account for this discrepancy later.
Y1 = mx+ b (Best fit line for YOUR data)
- Visually verify the line is a good match by doing a Zoom 9. Quit to your homescreen. Follow these keystrokes:
MATH 9 (fnInt) VARS Y-Vars 1(Function) 1 (Y1) Comma (,) X (variable key) Comma (,) lower limit value (0) Comma (,)
Upper limit value (from your data) ENTER
As we discussed above, even though this area is a triangle it can still be calculated as the length times width of many tiny rectangles placed side by side under the function. The result of this operation is the value of the integral of your force function.
4)How does the measured value of the work done on the spring from step 3 compare to the value you approximated in number 2?
Now we will perform a theoretical calculation of what energy should have been exchanged in this process for your spring. Mathematically we have already discussed that
as
Since F(x) for a spring is kx, according to Hooke’s Law, the expression becomes
where x’ is the maximum distance that you stretched the spring and k is your spring constant, determined from the slope of the graph in part I.
Integrating the linear function of x we get
or
This is an interesting solution if you recall that the potential energy stored by a stretched or compressed spring is
So the work done in stretching the spring in this problem results in energy stored by the spring. That energy can manifest itself as the elastic recoil of the spring, giving kinetic energy back to the mass that caused the compression in the system.
5)Using your values of k and x’, work out the integral shown above (don’t just jump to the end solution!) and get a numeric result. The number that you calculate will be the theoretical value of the work done by your spring.
6)Compare the theoretical value of the work done (step 5) to the actual value of the work done (step 3) and the approximate value of the work done (step 2). Describe how these numbers compare to each other.
7)Perform a % difference calculation on the actual and theoretical solutions from steps 3 and 5.
8)The actual and theoretical value should be exactly the same (and therefore a % difference of zero!). If yours are, great! Since there is probably some small discrepancy you need to account for this error. Identify some specific sources for this discrepancy.
9)Now examine your y-intercept value for your force function. Is it exactly zero? Should it be? Why?
10)Since your y-intercept is not exactly zero, explain how this affects the area you calculated that defines the work done by the spring.
11)How can you accommodate this difference in the computations of your actual work value? Propose a change for step number 3 to take into account this error.
12)Perform this new procedure and compare the result to the theoretical value of the work done by the spring (from step 5). Do a new % difference computation. The new value should be much smaller than the old one.
13)Repeat this method for calculating the work with the non-linear spring data from part I. Calculate both the actual work done by using the data and compare to a theoretical value using the function F(x) = Ax2. In evaluating the theoretical integral be sure to use your computed value of A from Part I step 19. Don’t forget to check the y-intercept!