STUDENT LEARNING OUTCOMES
Analyze a spring-mass system using Hooke’s law and a best-fit line.
Determine if a rubber band obeys Hooke’s law.
Combine the qualitative observations of a model with the quantitative results of an experiment into a coherent formal report.
EQUIPMENT
Camera
Paper
Paper clips
Paper cup
5+, set of coins (e.g. nickels, quarters)
Rubber band
Ruler
Sticky tape
String (dental floss works well).
WRITE-UP REQUIREMENT: FORMAL
Formatting – 4 pts, Introduction – 2 pts, Procedure – 3 pts, Data & Analysis – 7 pts, Conclusion – 4 pts
PART 1: HOOKE’S LAW SIMULATION
PREPARATION
As long as an ordinary spring is not stretched too far, the force it exerts is proportional to the distance it is stretched. This statement regarding the spring force is an application of the law of elasticity, which was discovered in 1660 by the English physicist Robert Hooke (1635-1703).
In mathematical terms, the magnitude of the spring force F is proportional to the displacement (∆x) from the equilibrium position, meaning Hooke’s law can be written as:
F “ ´k∆x (5.1)
The negative sign means the force is restorative. The proportionality constant k is called the spring constant (it also has other names such as the stretch modulus).
In this lab we will first simulate the behavior of a spring to observe idealized behavior. Then we will conduct the experiment using a rubber band. The data will be analyzed to determine if a rubber band behaves according to Hooke’s law.
PROCEDURE
(1) Open the following simulation which models an ideal mass-spring system: Masses-and-Springs (See Figure 5.1).
(2) From the main menu, select “Lab.”
(3) Throughout the experiment, keep the gravity slider set at 9.8 m{s2.
(4) Check the Displacement/Natural Length checkbox. This will mark the length of the spring with no mass attached.
(5) Click on the 100 g mass and attach it to the spring.
(6) The mass may oscillate up and down. You can use the stop button (shaped like a stop sign and located near the top of the spring) to stop this motion; we will not be studying this oscillatory motion yet.
(7) Record both the mass and the displacement in a table similar to Table 5.1.
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(8) Use the ruler tool to measure the displacement from equilibrium (∆x).
(9) From the mass used, find the downward force on the spring. (Consider what must the spring force be to keep the system in equilibrium.)
(10) Repeat this measurement for 4 other masses—for a total of five total data points. Figure 5.1: Simulation for a spring with a mass attached.
Table 5.1: Example data table.
Trial Mass (kg) Spring Force (N) Displacement (m)
1
2
3
4
5
ANALYSIS
Use your data to make a plot of ∆x versus F. Add a best-fit line, and include the equation of this line.
Considering Eq. 5.1, determine which quantity the slope of your best-fit line represents.
Using the slope of your best-fit line, determine the value of the spring constant, k.
PROCEDURE
The goal of this experiment is to determine if a rubber band obeys Hooke’s law (or not). You do not have to follow these directions exactly (this is just one way to do it); you only need to hang an adjustable mass from a rubber band and measure its displacement, much like in the simulation. One method of making the displacement measurements is shown in Figure 5.2. There are also some helpful hints listed at the end of this lab.
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Figure 5.2: Apparatus for testing if a rubber band obeys Hooke’s law. (left) Complete setup. (top-right) Example using a piece of paper as a mass holder. (bottom-right) Example using a paper cup as a mass holder and measuring the displacement.
(1) Find something to hang a mass from (doorknobs work well). The mass needs to hang close to a wall.
(2) Cut your rubber band so it no longer forms a loop.
(3) Tie a string to each end of the rubber band.
(4) Tie a paper clip to one end, and tie the other end to your hanger.
(5) For a mass holder, you may fold a piece of paper into an envelope, or attach a loop of string to the paper cup.
(6) Hang your mass holder from the paper clip.
(7) Add some initial mass to your mass hanger to get a little bit of tension in the rubber band. You do not need to know what this initial mass is.
(8) Tape a piece of paper to the wall, and neatly mark the equilibrium position of the mass hanger on the paper. This mark will be used as a reference for measuring displacement when additional weight is applied to the rubber band.
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(9) Prepare a table similar to the one shown above. If you are adding quarters for instance, change the first column to “Quarters added.” If you are adding a mix of coins, add more columns.
(10) Add some mass to your mass holder, and neatly mark the new equilibrium position.
(11) If the rubber band has not noticeably stretched, add some more mass—do not include any data points in which the displacement has not changed enough for you to measure.
(12) Use a ruler to measure the displacement for each mass added.
(13) Repeat these measurements for at least 8 different masses.
(14) Get a wide range of displacement values: be sure to get several where the rubber band has only stretched a little, a few where it is stretched almost to its limit, and a few in between.
Data collection table.
Coins added Mass added (kg) Force (N) Displacement (m)
1
2
3
…
Uncertainty estimate for displacement measurements: (m)
ANALYSIS
(1) Using your data, make a plot of ∆x versus F and add a linear best-fit line.
(2) Looking at your graph of the rubber band’s force vs. displacement, determine if a linear best fit line accurately represents the trend found in your data.
(3) Add a best-fit line to your plot, and make sure the equation is visible.
One method to evaluate how well your experimental data is represented by a linear best fit is to create a residual plot.
The residuals simply tell you far from the best-fit line the individual data points are. For example, if your data points all fall very close to the best-fit line, then all of your residuals will be close to zero. The residuals are calculated as the difference between a data point and the line of best fit. This is calculated as:
ri “ yi ´ ypxiq (5.2)
Where riis the residual for a given measurement, yiis the measurements, and ypxiq is the predicted value for your measurements. ypxiq is calculated from your best-fit line (i.e. Hooke’s law with your experimental results). To create a residual plot:
(4) In your spreadsheet, create a new column and enter the equation of the best fit line replacing ‘x’ with the cell containing the first independent variable.
(5) Drag the corner of this cell downward to auto fill the remaining cells, this will create a column of discrete points that fall along the best fit line.
(6) Create a new column in your spreadsheet (we will refer to this as residuals) and fill it with the difference between the column containing your dependent variable and best fit line.
(7) Create a scatter plot of the residuals versus the independent variable.
Consider the following for your report:
23
HOOKE’S LAW
How does the plot you obtained using a rubber band differ from the plot obtained using the “ideal” spring of the simulation?
Does your residual plot display any trends? For example, are there sections of this plot where the data tends to be negative or positive?
What do these results tell you about the validity of applying a linear fit to the data?
Based on your response to the previous questions, does a rubber band obey Hooke’s law?
