Honors Physics Energy& SHM Lab

The purpose of this investigation is to verify the concepts of Conservation of Energy and Simple Harmonic Motion.  This will be accomplished by hanging a mass from a spring and allowing it to oscillate vertically.  In the absence of nonconservative forces such as friction, the energy of the mass-spring system should remain constant.  Force will be measured by a force sensor from which the spring hangs.  Time and position will be measured by a Calculator Based Ranger (CBR) connected via LabPro Interface to a laptop computer.  The program Logger Pro will interpret and graph the data and allows for the calculation of the elastic, gravitational, and kinetic energy of the mass-spring system.

Procedure

Set two ring stands on a lab table.  Attach a force sensor to one end of a long metal rod and support it between the two ring stands as shown in the diagram.    The ring stands should be near each end of the metal rod.  The CBR will be placed on the floor directly below the force sensor.  A spring will be hung from the force sensor and a mass from the bottom of the spring.  IMPORTANT:  Great care must be taken not to drop a mass on the CBR!!  It would be a good idea to always move the CBR to a safe location and ONLY put it below the mass during the very brief interval of time when it is actually in operation!

1. Connect the LabPro to a USB port on the computer, connect the CBR to LabPro DIG/SONIC 1 and the Force sensor to LabPro CH1, and plug in the power supply for the LabPro.  The LabPro does not have a power switch.  Run the program Logger Pro 3 using the file named SHM Energy.

2. Calibrate the force sensor.  Go to the Experiment menu and then Calibrate and choose the force sensor. You will enter two values to calibrate the sensor.  Read the next two steps before you proceed with the calibration.

3. Hang only the spring from the sensor.  Click on the button Calibrate Now to perform the calibration.  Then enter zero as Reading 1 and click Keep.  This means that the computer will record zero newtons for force when the spring is in this condition and it is not exerting any force on the test object.

4. Hang the test object on the end of the spring and find the position at which it will rest without accelerating.  For Reading 2, use the mass to calculate and enter the correct weight (in newtons!) of the test object (not including the spring!) and click Keep.  This means that the computer will record a force equal to the weight of the test object when the spring is in this condition and is exerting that much force on the test object.  All other readings from the sensor are now adjusted according to the two known force values that have been entered by you – this is done by the program.

5. Check that all of the support structure is very secure.  The bobbing mass should be about 50 to 60 cm above the location where the CBR will rest. 

6. Carefully lift the mass upward and let it go so that it bobs up and down with an amplitude of around 10 cm.

7. Once the mass is already in motion and oscillating smoothly, place the CBR directly below it and click on the Collect button.

8. This should produce smooth graphs showing fluctuations of position and force with respect to time.  If there are any significant glitches in these graphs then make adjustments and repeat the experiment.

9. Remove the CBR to a safe location.

10. Use the Save As command to save to the USB drive your data in a file with the last name of one of the persons in your group.  Feel free to Save as often as you like so that you could easily recover a previous step if you make mistakes.

Analyses

Note:  You have complete control over the appearance of graphs – single click, double click, or right click when you want to change something!  Or try the Options menu.  To change colors or symbols for the data double click the column heading in the data table or try the Data menu, Column Options…

Position vs. Time, Force vs. Acceleration, and Force vs. Position:

1. The program has multiple pages with various graphs.  Page 1 shows the Data Table and small versions of the graphs of Position and Force vs. time.  This page gives you a quick view of the results of the experiment. 

2. Go to Page 3 on which you should find a full-sized Position vs. Time graph.  Do a regression using a sine model.  (click Analyze, Curve Fit ...)  If you have done things correctly you should be able to do the regression over the entire graph using all data.  (In other words you should not have to select a portion of the data.)  Adjust the appearance if necessary – check for appropriate labels, units, scales, etc.  Use point protectors but do not connect the points.  Under the File Menu use Print Graph to print the graph.  In the comment section enter the names of the members of your group.  Return the graph to its original size.

3. On Page 4 find the Force vs. Position graph.  Do a regression using a linear model.  Adjust the appearance if necessary – check for appropriate labels, units, scales, etc.  Use point protectors but do not connect the points.  Use Print Graph to print the graph.  Note:  you will need to use the equation from this graph in order to correct the equation for calculating elastic potential energy – this is explained below.

4. On Page 5 find the Force vs. Acceleration graph.  Do a regression using a linear model.  Adjust the appearance as before.  Use Print Graph to print the graph. 

Energy vs. Time, Energy vs. Position, and Data Table:

5. You must enter the mass of the test object.  To do this use, go to the Data menu, then Column Options – Mass.  Under the definition tab choose Generate Values using the Numeric Fill option.  Enter your mass as the value and 61 as the number of rows.

6. You will need to modify the equation for gravitational potential energy.  To do this use, go to the Data menu, then Modify Column – gravitational potential energy.  Under the Definition tab you should see an equation that is used by the computer to calculate the value.  An appropriate type of equation has been entered but you will need to modify the equation to correctly describe your particular experiment.  We will use the lowest point in the object’s motion as a reference for gravitational potential energy.  Use the position vs. time graph to find the lowest position and then enter it into the equation.  For example if the lowest position of the mass was 0.65 m, you would make the formula read:  “Mass”*9.8*(“Position”−0.65).  Make the necessary changes and click OK.

7. You will also need to modify the equation for elastic potential energy.  Use similar steps to the ones taken in the previous step.  The equation should be:  0.5*k*elongation^2.  The value of k can be found from the force vs. position linear regression.  The elongation is the difference in the position of the mass as it moved and the position at which the force of the spring would have been zero.  Use the linear regression from force vs. position to solve for the position at which the force of the spring is zero.  For example, if the linear regression indicates k = 25.0 N/m and the force equals zero at a position of 0.85 m, then you would enter the equation:  0.5*25*(0.85−“Position”)^2.

8. Check the equations entered for kinetic energy and total energy and make sure that these are correct.

9. On Page 6 find a graph of Energy vs. Time.  Make sure there is a legend on the graph.  Use point protectors and make sure all of the plots have different symbols.  Choose to connect the points.  There will be no regression equations on this graph.  Instead, do a statistical analysis of the total energy by clicking on the Stat button in the tool bar and choosing ONLY the total energy.  Adjust the appearance if necessary – for example, the legend and the statistics boxes should not cover significant features of the graph.  Use Print Graph to print this graph.

10. On Page 7 find the Energy vs. Position graph.  Adjust as necessary and then use Print Graph to print the graph.  Note: you do not need to repeat the statistics on this graph because it will be the same result.

11. Use Print Data Table to print the Data Table found on page 2.  Note: choose to print only from page 1 to 1 – do not print all pages (you will not need all of the data for your lab report – one page should be sufficient).

12. Before turning everything off check to see that you have all necessary graphs and Save the file one last time.

Questions

1. Highlight or otherwise indicate one row from the data table and check the computer’s calculations by doing them yourself:  calculate kinetic energy, gravitational potential energy, elastic potential energy, and total energy.  Show work.

2. Discuss whether or not your results support Conservation of Energy and explain how so.  Refer specifically to one or more graphs.

3. Discuss whether or not your results support the concepts of Simple Harmonic Motion: 
(a) Refer to the Force vs. Position graph and describe specifically whether it shows that the object was under the conditions that cause SHM. 
(b) Refer to the Position vs. Time graph and describe specifically whether it shows the object exhibited the expected type of motion associated with SHM.

4. Show work as you:
(a) Use the Position vs. Time graph to determine the period of the object’s motion. 
(b) Use your values for k and m to calculate the theoretical period. 
(c) Calculate the percent error, taking the first value to be the “true” period.

5. Consider the regression equation of the Force vs. Acceleration graph: 
(a) Explain how it is consistent (or not) with Newton’s Laws. 
(b) What is the significance of the slope?  the y-intercept? 
(c) Make any appropriate calculation(s) such as error or deviation that will quantify the accuracy and/or precision of the slope and/or y-intercept.

6. Use the statistical results on the Energy vs. Time graph to determine the percent deviation in the total energy of the mass-spring system.

7. Write an intelligent, grammatically correct paragraph or two discussing both the signs of error and the likely sources thereof.

A complete report (50 pts):  (pages in this order)

q Data Table  (6)

q Position vs. Time graph, with regression equation.  (6)

q Force vs. Position graph, with regression equation.  (6)

q Force vs. Acceleration graph, with regression equation.  (6)

q Energy vs. Time graph, with statistical analysis of total energy  (6)

q Energy vs. Position graph  (6)

q On separate paper, responses to the questions.  (14)