Measuring the Spring Constant k
In this experiment, you will measure the spring constant of a given spring. Granted, that alone doesn’t make for a very exciting lab. However, the way you do it today will introduce you to an essential tool for any scientist: curve-fitting!
Here’s how it will work. Hang your given spring from a stand (you can find these materials in the cabinets on the south side of the room). Mount a ruler along side of the spring, and note where the end of the spring is. Hang a mass at the end of the spring, and note how far the end of the spring has stretched. Do this with at least 10 different masses. MAKE SURE YOU DO NOT ADD SO MUCH MASS TO YOUR SPRING THAT IT STRETCHES OUT (is stretched beyond its elastic limit)!!!!!!
Here’s a free body diagram on a hanging mass:
ΣFx = mg – kx = 0, if mass just hangs without moving. Thus
mg = kx or m = (k/g) x
Note that if you plot your mass (m) on the y-axis and your distance stretched (x) on the x-axis, your data should follow a straight line with a slope of k/g.
Get a laptop and log on. On our course web site is an Excel spreadsheet called “LinFit-1a.xlsm”. Open this up.
1. Type in your distances stretched (in meters!) in the column marked “x,” and the uncertainties in these distances in the column marked σx.
2. Type in your corresponding masses (in kg!) in the column marked “y,” and the uncertainties in these masses in the column marked σx. It is a good idea to check your masses by weighing them—many of them can be different from what they say they are by a few grams.
3. You should now see all of your data points plotted automatically with error bars. It should look something like the figure below. If there are some data points or uncertainty bars missing, get your prof to help you.
PLEASE CHANGE THE LIMITS ON YOUR GRAPH SO THAT THE DATA FILLS THE GRAPH AREA! I don’t want to see all your data points crammed into one corner of your plot.
4. Run the macro. To do this, click on the View tab and click on the “Macros” button at the far right. Run the 'LinFit-1a.xlsm'!LinFit.LinFit macro (not the “test1” macro) and do NOT force your y-intercept to be zero when prompted (hit “No”). What this does is it finds the slope and y-intercept of the line that best fits your data. These values are displayed just above your graph, and are presented like what is seen below:
y uncertainties onlysai / Red C2
a0 / -294.16284 / 0.418381658 / 4.594181
a1 / 3.53374517 / 0.003534675
With x and y uncertainties / # of Iterations / 11
sai / Red C2
a0 / -294.16284 / 0.64646179 / 1.924277
a1 / 3.53374517 / 0.005461598
The numbers you care about are the ones marked “With x and y uncertainties”. In the above example, the value for a1 (3.53374517) is the slope of your graph (equal to k/g), and the uncertainty in the slope is 0.005461598. Use these numbers in your analysis. The values for a0 are the numbers for the y-intercept and its uncertainty. Take note of your “Red X2” (reduced χ2) value (above, this is 1.924277). If this is greater than 5, get your prof for help.
5. Use these values for the slope and the uncertainty in the slope to determine your value for k and the uncertainty in k. Since slope = k/g, then k = g*slope. You get the uncertainty in k the usual way (maximize, etc.).
Your report will contain the usual stuff.
Name/Partners/Date/Title
Introduction – State the purpose of the experiment.
Theory – Show how the slope in this graph gives you the spring constant.
Experiment – Describe your experiment as though this “manual” doesn’t exist. Don’t forget equipment numbers and a diagram. Include all of your data here.
Analysis – Include your graph here, and your calculation for k and the uncertainty in k.
Discussion – Is this a reasonable result? Why or why not? If it is not reasonable, go back and check your work! Don’t turn in flawed results. Also answer the question: Why isn’t the y-intercept of your graph zero?
Conclusion – Restate your value for k and its uncertainty. Say again why you think this is reasonable. Nothing new should go here.
Richard A. Thomas – UST Physics