Goals

The goals of this simulated lab exercise are to a) show that a rubber band (simulated) obeys Hooke's Law and b) to determine the spring constant of the rubber band to 3 significant figures.

 Introduction & Theory

A spring scale simply uses the tension of a spring to measure force. You can make your own spring scale with a spring if you happen to have one lying around. It's even easier to use a large rubber band. In order to make it function like a measuring instrument, you have to calibrate it.That is, you have to know how much force corresponds to how much stretch or elongation of the band. You may ask, "How can I know how many newtons of force it takes to stretch the band a certain amount if I don't have a calibrated scale to begin with?"  Well, one possibility is to hang calibrated masses from the rubber band. If, for example, you have lab masses calibrated in grams, then you know how to calculate the weight of the mass in newtons. If you don't have calibrated masses, you could just as well use a number of objects of identical mass. For example, you could suspend washers from the band and see how much additional stretch you get for each washer added. You would then have a rubber band calibrated to measure force in units of washer weights. That would be sufficient for comparing magnitudes of forces. All measurements, in fact, are comparisons to a standard. The only thing special about a newton is that the worldwide scientific community has agreed to measure force in newtons rather than, say, washers.

An important characteristic of a spring or rubber band scale is that it be linear. Suppose a force is applied to stretch the spring. The spring, of course, pulls back with a tension force. Suppose that when the spring exerts a force T, the spring is stretched a distance L beyond its unstretched position. If the spring is stretched further, say to a distance of 2L, the spring exerts a tension force twice as much or 2T. If in general the tension force is proportional to the stretch, the spring is said to obey Hooke's Law. Many springs obey Hooke's Law as long as you don't stretch them too far. The same is true for rubber bands. Let's examine a specific situation.

Suppose you hang a rubber band vertically and then hang a platform from the band as shown in Figure A below. The purpose of the platform is to hold weights. You start with the platform hanging freely at rest in what's called its equilibrium position. Let's say you denote the bottom of the platform as the equilibrium position. This is an arbitrary but convenient designation, as any point of the platform will do.You then add a weight to the platform and it stretches as shown in Figure B.

The force diagrams for the two situations are shown below the corresponding pictures. As long as the platform is in equilibrium, the net force on the platform will be 0 and the magnitudes of the tension and weight forces will be equal. However, the magnitudes of both forces will be greater with an object placed on the platform than without.

Figure A	Figure B

For a spring (or rubber band or other elastic medium) that obeys Hooke's Law, the difference in tension forces, (T₂ - T₁), is proportional to the difference in positions, (x₂ - x₁). In equation form, we write

(T₂ - T₁) = -k(x₂ - x₁),          [Eq. 1]

where k is the constant of proportionality. We call this special constant the spring constant. Note that k has units of N/m (or kg/s²) and is always positive. The negative sign in front of k in Eq. 1 is needed, because the change in the tension force is always opposite the displacement from equilibrium. If the elastic medium is stretched downward from equilibrium, the tension force increases in the upward positive direction while the displacement increases in the negative direction. If the medium is compressed upward from equilibrium, the tension force increases in the downward negative direction while the displacement increases in the positive direction.

We can use the Δ notation to represent the difference of initial and final quantities and express the relationship more compactly as ΔT = -kΔx. In words, this relationship says that the change in tension force ΔT on an object is proportional to the resulting displacement of the object and of the opposite sign.

An application of Hooke's Law to a system in equilibrium: For the special case described of a weight hanging from a rubber band in equilibrium, we can make the substitution (T₂ - T₁) = (W₂ - W₁) in order to obtain

(W₂ - W₁) = -k(x₂ - x₁).         [Eq. 2]

The difference, (W₂ - W₁), is just the weight of the object that was added to the platform. The difference, (x₂ - x₁), is the displacement that results from the addition of weight. In compact notation, ΔW = -kΔx.

Given that Eq. 2 is valid for the equilibrium situation, here's how one would use it to determine the spring constant of a spring or rubber band.

1. Measure the position of the weight hanger for several values of weight.
2. Plot a graph of the displacement of the weight hanger vs. the weight added to the weight hanger.
3. If the graph is linear, the spring obeys Hooke's Law.
4. Determine the spring constant from the appropriate coeffiicient of the fit.

 Setting up Logger Pro

You'll be able to show all your work for this exercise in a Logger Pro file. Open up LP now.  Set up the Data Table as follows.

1. Double click on the X column in the data table. 
2. Change the name to Added Mass, no shorthand name, and units of kg. Note that the Added Mass does not include the mass of the platform.
3. Click on the Options tab and select a Displayed Precision of 3 decimal places. 
4. Now double click on the Y column. 
5. Change the name to Position, no shorthand name, units to m, and Displayed Precision to 3 decimal places. 
6. You'll need a 3rd column to calculate the displacement from the original position. On the menu bar, click Data --> New Manual Column. 
7. Enter name of Displacement, shorthand name of Delta-x, and units of m.  Set the Displayed Precision appropriately.
8. You'll need a 4th column to calculate the added weight corresponding to the added mass. On the menu bar, click Data --> New Calculated Column. 
9. Enter name of Added Weight, shorthand name of Delta-W, and units of N  For the equation, click first on Variables and select "Added Mass". 
10. Now, in the Equation window, append *9.8 so that you end up with this:  "Added Mass"*9.8.  This will have the effect of multiplying the entries in the Added Mass column by 9.8 in order to obtain force in newtons.

Next set up the graph. 

11. Decide which variable is independent and which is dependent. Then plot the variables on the appropriate axes according to convention.
12. Now double click anywhere on the graph to bring up the Graph Options. Type the appropriate title. 
13. As always, unselect Connect Points and select Point Protectors if these aren't selected by default.
14. Click on Options, and set both axes to Autoscale from 0.

Save your LP file using the name M04-lastnamefirstinitial. Now you're ready to enter data.

 Taking data

You'll take data from an IWP applet. Click here to open the applet. Read the problem description completely. You'll have to scroll the description window up in order to see all the text.

1. Begin by reading the position labeled Unstretched position. Record this value in the Notes box of your LP file. Note that all positions are negative since +y is up.
2. You'll read positions as a function of the added mass. It's important to consistently read positions from the same part of the platform. Use the bottom of the platform for this.
3. Run the applet to release the red stick holding the mass in place. Let the motion damp down to equilibrium. You can increase the damping coefficient so that the motion will damp quicker. Once the platform stops, read the position (bottom of the platform again) and enter into your LP data file. Note that the Added Mass for this data point is 0.
4. Now add mass in increments of 0.100 kg, let the platform settle down to equilibrium, and read the position. Continue to take readings up to a total Added Mass of 0.500 kg.
5. Calculate the Displacements and enter them in your data table. Remember that all displacements are calculated from the position for which Added Mass = 0. Note also that since +y is up, displacements due to the added weights will be negative.
6. Your graph should have been plotted automatically. If there are lines connecting the points, you weren't paying attention when we warned you about that above. Remove any connecting lines now. Connecting the dots is something you do in grade school. In physics, you construct a best fit line. That's next.

 Analysis: Fitting the data

1. We expect the variables to exhibit a linear relationship. On the menu bar, select Analyze --> Linear Fit. The best-fit line will be drawn and the fit results displayed.
2. Right click on the box that displays the fit results. Select Linear Fit Helper Options. Then select the appropriate Displayed Precision. 
3. In the Notes window, prepare the matching table. You won't have an expected value for the slope but you should know what to expect for the intercept.
4. Write the equation of the fit in the Notes window.

 Interpretation

1. Save your file and upload it to the first item of WebAssign M04.
2. Write answers to the following in the second item of WebAssign M04.
   1. Using one of the coefficients of the fit, determine the spring constant of the rubber band. Do not use a data point for this. Show your work.
   2. Why is 0 the expected value for the intercept?
   3. Use your equation of fit and any other information you need to calculate the mass of the platform. Show your work clearly.

 Submission

Submit WebAssign M04 by the due date.

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