Lab
Static Springs: LabPro Data for Hooke's Law
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Purpose
The purpose of this lab is to use linear regression and data analysis to calculate the elasticity constant for a single spring, and for springs arranged in series and in parallel.
Equipment
two springs
slotted masses
mass hanger
trapeze
LabPro
meter stick/ruler (optional)
Procedure and Data Tables
Part I: Single springs
. Complete ten trials for each spring. Before suspending each spring measure its equilibrium length.
Suspend the spring with its empty mass hanger from the trapeze and align the distance detector directly underneath. Run LoggerPro (Start, Programs, Math, LoggerPro 3.1) and use the data table on the left side to approximate, to three decimal places, how far the bottom of the mass hanger is suspended above the motion detector. Record this answer in your data table. Then increase the hanging mass by increments of 100 grams and record each subsequent stretched position.
Continue until you have suspended a total of 950 grams from each spring. Please note that the empty mass hanger has a mass of 50 grams. Complete the following data table for each individual spring.
At no time during each phase of this experiment are you allowed to change the height of the suspension point on the trapeze. If the bottom of your mass hanger comes closer than 40 cm above the motion detector, it will no longer provide correct readings.
single spring #1
Equilibrium position
_____________
Trial
Total Hanging Mass
Total force (mg)
Stretched position
1
50 g
2
150 g
3
250 g
4
350 g
5
450 g
6
550 g
7
650 g
8
750 g
9
850 g
10
950 g
single spring #2
Equilibrium position
_____________
Trial
Total Hanging Mass
Total force (mg)
Stretched position
1
50 g
2
150 g
3
250 g
4
350 g
5
450 g
6
550 g
7
650 g
8
750 g
9
850 g
10
950 g
Part II: Springs in Series
. Complete five trials using both springs arranged in series by linking them together one below the other. Initially record the equilibrium length of the springs without any suspended masses. Complete five trials ranging from 50 grams to 450 grams.
springs in series
Equilibrium position
_____________
Trial
Total Hanging Mass
Total force (mg)
Stretched position
1
50 g
2
150 g
3
250 g
4
350 g
5
450 g
Part III: Springs in Parallel
. Complete five trials using both springs arranged side-by-side, or in parallel. Initially record the equilibrium length of the springs without any suspended masses. Complete five trials ranging from 150 grams to 950 grams. Be careful to make sure that the suspended masses are centered so that they are supported equally by both springs.
springs in parallel
Equilibrium position
_____________
Trial
Total Hanging Mass
Total force (mg)
Stretched position
1
150 g
2
350 g
3
550 g
4
750 g
5
950 g
Data Analysis
EXCEL will now graph your data. Minimize your browser, double click My Computer, double click the shared drive called colwell/bay on Lederman, double click your period's folder and then finally double click 1-
HookesLawLabPro.xls
. You will most likely be asked to open the file as "read only" - that is fine. As soon as the file is open, use File Save As to rename the file as
LastnameLastnameHookesLawLabPro.xls
in your period's folder. This copy of the file now belongs uniquely to your group. Remember that there are to be no spaces in the file name.
What is the name of your file?
Once you save your file with all of the information for each of the individual springs as well as the two special arrangements, print out a copy of your file for your lab reports. Don't forget to rescale any axes to insure that you maximize the display of your data.
On your printout, use the form y = mx + b to rewrite the equation of each graph next to the graph's title. Remember to use the variables
x
and
F
.
What is the numerical value of the slope of the individual graph of spring #1 in m/N?
What is the numerical value of the slope of the individual graph of spring #2 in m/N?
Hooke's Law
When working with springs,
Hooke's Law
states that
F
_{external}
= ks
where
F
_{external}
is the applied force,
k
is the spring's elasticity constant measured in N/m, and
s
is the displacement, or how far the spring is stretched from its equilibrium position.
Note that our graphs have the applied force (the suspended weights) displayed on the x-axis and the position of the mass hanger above the motion detector along the y-axis. This is because we manipulated the values of the suspended masses, resulting in force or weight being our independent variable while we merely recorded the stretched positions as our experimental outcomes, or dependent values.
Since our graphs are
position vs Force
, the elasticity constant for each of our spring systems will be calculated as the negative reciprocal of each graph's slope.
What is the elasticity constant (k
_{1}
) for spring #1?
What is the elasticity constant (k
_{2}
) for spring #2?
Springs in Series
We will now investigate the elasticity constant for springs arranged in series. The theoretical constant for such a system of springs is given by the formula:
What is the slope of your third graph when your springs were arranged in series?
Based on your 3rd graph's slope, what is the
experimental elasticity constant
when your springs were arranged in series?
Use the original elasticity constants for spring #1 and spring #2 and the formula shown above for springs in series to calculate the
theoretical value
for the elasticity constant when your springs were arranged in series? (Show your calculations next to this graph.)
What is your percent error for this phase of the experiment? (Show your calculations next to this graph.)
Springs in Parallel
We will now investigate the elasticity constant for springs arranged in parallel. The theoretical constant for a system of springs in parallel is given by the formula:
What is the slope of your fourth graph when your springs were arranged in parallel?
Based on this graph's slope, what is the
experimental elasticity constant
for when your springs were arranged in parallel?
Use the original elasticity constants for spring #1 and spring #2 and the formula shown above for springs in parallel calculate the
theoretical value
for the elasticity constant when your springs were arranged in parallel? (Show your calculations next to this graph.)
What is your percent error for this phase of the experiment? (Show your calculations next to this graph.)
Summary
When springs are connected in series, the spring constant of the system ____ when compared with the elasticity constant for either of the original springs:
increased
decreased
remained the same
When springs are connected in parallel, the spring constant of the system ____ when compared with the elasticity constant for either of the original springs:
increased
decreased
remained the same
Inverse Functions
State the inverse function of Spring #1 equation; i.e, instead of stretch = force(slope) + constant, write it as force = .....
State the inverse function of Spring #2 equation; i.e, instead of stretch = force(slope) + constant, write it as force = .....
State the inverse function of the equation for your springs when in series; i.e, instead of stretch = force(slope) + constant, write it as force = .....
State the inverse function of the equation for your springs when in parallel; i.e, instead of stretch = force(slope) + constant, write it as force = .....
Energy and Work
Calculate the work done (and consequently, the energy stored) in Spring #1 if it were to be stretched from equilibirum to 0.03 meters.
Calculate the work done (and consequently, the energy stored) in Spring #2 if it were to be stretched from equilibirum to 0.03 meters.
Calculate the work done (and consequently, the energy stored) when the springs are arranged in series and stretched from equilibirum to 0.03 meters.
Calculate the work done (and consequently, the energy stored) when the springs are arranged in parallel and stretched from equilibirum to 0.03 meters.
What is the relationship between the work done when the springs are arranged in parallel and when stretched individually?
What is the relationship between the work done when the springs are arranged in series and when stretched individually?
After submitting your results online, turn in your written lab report to the one-way box. Each report should include a cover page, a copy of your data charts, and a well-documented printout of your graphs.
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Catharine H. Colwell
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Application Programmer
Mark Acton