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.
A simple pendulum consists of a small object (the “bob”) suspended by a lightweight cord. The mass of the pendulum is actually only the mass of the bob; the mass of the string is not included. The period of a pendulum is the amount of time for the bob to complete exactly one cycle or oscillation back and forth. The length of the pendulum extends from the attached end of the string to the center of mass of the bob. The amplitude of 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. 309 – 320 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, and g = a gravitational constant
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 swinging before you 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. Lab reports are not a group project. Each group is to complete Part A and then choose either Part B or Part C (but not both).
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, and k (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 is important that the amplitude of the bob’s oscillation remain less than or equal to 10° - the 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 collect one more datum with a length of 5.0 cm. This will give you a total of eleven rows of data in your table.
Choose only one of the following:
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 is important that the amplitude of the bob’s oscillation remain less than or equal to 10° - the smaller the better. Start with the mass of your original bob. Then attach bobs of various masses and measure the resulting period of each.
Measure and record the length and mass of the bob and keep both constant. Make a data table with 4 columns: amplitude, time, # of cycles, and period. In this part of the experiment you will vary the size of the bob’s oscillation and measure the resulting period (without changing the length or the mass). Start at 10° and increase to 90° in increments of 10°. This will give you a total of nine rows of data in your table. Note: As you attempt to measure greater and greater amplitudes you may find it difficult to complete 10 cycles as you time the period. In such a case you may have to use a lesser number of cycles in your timing – use your best judgment and some common sense.
1. For each part of the lab performed (part A and part B or C) 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: For this first lab report I want
you to do the graphical analyses “by hand” – in other words do not use a
graphing calculator to find the best fitting equations. I expect to see work showing how you
determined each equation.
1. How do your graphs and results support or refute the theory of a simple pendulum? Your purpose here is to discuss, 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. (3)
2. Your work with the pendulum is an indirect way to measure g, an important constant related to the strength of gravity. The accepted value for g is 9.80 m/s^{2}. (a) Calculate the value of g based on your results using the equation: g = 4π^{2}/k^{2}, where k = 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 as k). (c) Find the relative error for each of these two values of g. (2 ea)
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 of g
= 4π^{2}/k^{2}.
Put another way I am asking you to show how to combine the two equations
into one single equation that relates g and k. 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 and in this order:
Part A data
table (4)
Part A graph –
Period vs. Length (with best fit and equation)
(8)
One of B or C: (12)
Part B data table
Part B graph – Period vs. Mass (with best
fit and equation)
or
Part C data table
Part C graph – Period vs. Amplitude (with
best fit and equation)
Data table – Period
vs. Square Root of Length (4)
Graph – Period vs.
Square Root of Length (with best fit and equation) (8)
Answers to
Questions 1 – 4 (point values given
above)