The Pendulum Lab
Purpose
The purpose of this lab is to investigate and verify (or refute) the theoretical relationships among period, length, mass, and amplitude of a pendulum by graphical analysis.
Theory and Definitions
A simple pendulum consists of a small object (the “bob”) suspended by a lightweight cord.Themassof the pendulum is actually only the mass of the bob; the mass of the string is not included.Theperiodof a pendulum is the amount of time for the bob to complete exactly one cycle or oscillation back and forth.Thelengthof the pendulum extends from the attached end of the string to the center of mass of the bob.Theamplitudeof the pendulum is the angle formed by a vertical line and the cord when the bob reaches its maximum outward displacement.
As long as it does not swing too far from center, a simple pendulum exhibits a special type of behavior known in physics as simple harmonic motion (SHM).See pp. 296-306 of your book for more details about SHM.According to theory, the period of a simple pendulum will not depend on the mass of the bob or the amplitude (so long as it is less than about 10).Instead, the period will depend only upon the length as indicated in the following equation:
where:T= period,l= length, andg= a gravitational constant
Procedural Notes
Suspend the bob from a ring stand using string.Rigidly attach a meter stick to the ring stand such that the zero on the scale is aligned with the point where the string is attached.When measuring the length read the value at the point where the bob just touches the meter stick (which should be at the center of the bob).Attach a protractor to the meter stick such that the vertex is aligned with the point where the string is attached.When determining the period of the pendulum measure the total time required for a certain number of cycles and then divide by that number.In most cases it is convenient and sufficient to measure ten cycles.The greater the number of cycles that you measure the greater should be the accuracy and precision of the calculated period.Let the bob start swingingbeforeyou do the timing of the ten cycles; in other words do not try to release the bob and start the stopwatch simultaneously.
You will collect data as a group and each person in your group will use the same data to prepare the lab report.However, everything in your report except the data should be your own unique work.Labreportsarenota group project.Each group is to complete Part A and then choose either Part B or Part C (but not both).
A.Period vs. Length
Measure and record the mass of the bob.Record the amplitude as less than or equal to 10.Make a data table with 5 columns:length, time, # of cycles, period, andk(the constant in the best fit equation – to be completed later during your analysis).In this part of the experiment you will vary the length of the pendulum and measure the resulting period (without changing the mass or amplitude).It isimportantthat the amplitude of the bob’s oscillation remainless than or equalto 10the smaller the better.Start with a length of 100.0 cm and decrease in steps of 10 cm until you reach a length of 10.0 cm.Then collectone moredatum with a length of 5.0 cm.This will give you a total of eleven rows of data in your table.
B. Period vs. Mass
Measure and record the length of the bob and keep it at a constant value (try around 25 cm).Record the amplitude as less than or equal to 10.Make a data table with 4 columns:mass, time, # of cycles, and period.In this part of the experiment you will vary the mass of the bob and measure the resulting period (without changing the length or amplitude).It isimportantthat the amplitude of the bob’s oscillation remainless than or equalto 10the smaller the better.Start with the mass of your original bob.Then attach bobs of various masses and measure the resulting period of each.
Interpretations of the Data (Analyses)
1.For each part of the lab performed (part A and part B) make an appropriate and well-labeled graph.On each graph draw the best-fit line or curve and determine the equation that best represents the relation between the variables.Show all work on the graph itself.Remember you must include units in all work!
2.Using the data collected in part A, prepare a table of period and square root of the length.Make a graph of period vs. square root of length.Consider it to be linear, draw the best line, and determine the equation.As always show all work on the graph.
Note:You may find the best fitting equation of the line by doing the graphical analyses by graphing calculator or “by hand” If you do the work “by hand” I expect to see work showing how you determined each equation.
Questions(Write responses on separate paper!Questions 1 and 4 require complete sentences.)
1.How do your graphs and results supportorrefute the theory of a simple pendulum?Your purpose here is todiscuss, in writing, what evidence there is that indicates (or does not indicate) that the theory is correct.Do not do any calculations for this question.Instead, refer to the shapes, features, and types of relations and equations shown on the graph(s).Be sure to discuss each part of the lab.(2)
2.Your work with the pendulum is an indirect way to measureg, an important constant related to the strength of gravity.The accepted value forgis 9.80 m/s2.(a) Calculate the value ofgbased on your results using the equation:g= 4π2/k2, wherek= the constant derived from your data in part A.(b) Repeat using the slope form the period vs. square root of length graph (which should be about the same value ask).(c) Find the relative error for each of these two values ofg.(2 ea = 6pts)
3.So where did the above equation come from?!Figure this out for yourself by deriving it from the following two equations:
and
Copy these two equations on your paper and then show the algebraic steps that will result in a derivation ofg= 4π2/k2.Put another way I am asking you to show how to combine the two equations into one single equation that relatesgandk.Do not do any numerical calculations for this question.(2)
4.Write an intelligent, grammatically correct, and concise discussion of error in this lab.Any complete discussion of error will include:indication or evidence of error(s) and speculation or explanation of the most likely sources of the same.Remember to consider both types of error – random and systematic.Your goal here is to satisfactorily explain how and why your results are not perfect.(3)
A complete report (50 pts.) shall consist of the following– neatly labeled andin this order:
Name, Date, Block(1 pt)
data table(4 pts)
graph (with best fit & eq.)(8 pts)
data table(4 pts)
graph (with best fit & eq.)(8 pts)
data table(4 pts)
graph (with best fit & eq.)(8 pts)
Answers to questions(13 pts)