Lab 7: The Pendulum Lab
Objective:
1) Determine whether the period of a pendulum depend on the initial angle of
release (i.e. the initial amplitude).
2) Use a pendulum to measure “g” assuming it exhibits simple harmonic motion.
Lab Style: Cookbook
Definitions:
T = period, or the length of time it takes for the pendulum to swing back and forth
once.
Bob = the mass at the end of the pendulum’s string.
L = length of the pendulum, measured from the post to the center of the bob.
g = acceleration due to Earth’s gravity
Experimental System:
A spherical mass is tied to a string; the other end of the string is tied to a pivot
point. The mass is let go and allowed to swing freely.
Part 1: Angle Dependence
Outline: We will be varying the angle the pendulum is released from and determining
if this initial angle has an effect on the period. We will be measuring the following
angles:
θ
6
10
15
25
35
45
Measurement
1) Lay the bob and meter stick on the table. Measure out 60 cm of pendulum,
starting at the center of the bob. Record this value in your lab notebook. Carefully
tie the string to the post on the protractor at the point you just measured.
2) One student should displace the bob until the string measures out to the desired
angle on the right. The other student should get ready with a stopwatch and stand
directly in front of the pendulum’s rest position.
3) The student holding the bob should GENTLY let go. The student with the stop
watch should start measuring the period:
a. Start the stopwatch when the pendulum first passes through zero degrees
going from right to left. This is swing zero.
b. Keep track of how many times the pendulum passes through zero degrees
going right to left.
DO NOT TRACK PASSES THROUGH ZERO FROM LEFT TO
RIGHT.
c. When then pendulum passes through zero from right to left for the 5th
time, stop the stopwatch. Record this value in a table in your lab notebook
in pen.
4) Measure the period five times for each angle; enter the data into your table.
Analysis
1) Because you measured 5 swings instead of one, the data you recorded is 5T
instead of T. Divide your data by 5 to get the values for T and enter them into
your chart.
2) Find the average period, T , for each angle. Enter this in your chart.
3) Find the sd of T for each angle. Enter this in your chart.
4) Find the sdm of T for each angle. Enter this in your chart.
5) Make a graph (at least 1/2 page) that is labeled with the angles along the x axis
!
and time along the y-axis. Plot your T for each data point. Now put error bars on
the plot (SDM). Also, indicate on the plot what the theoretical value of T (within
the simple harmonic approximation) is by drawing a dashed horizontal line that
intersects that point on the y-axis.
!
Question: Within your experimental uncertainties, do you conclude that the period
does or does not depend on the initial angle? If it does, what is the percentage
difference between the periods for 6o and 45o?
Part 2: Length Dependence of the Period and measurement of g
Outline: In this section we will use the period of a pendulum to measure the value of
g. We assume the pendulum exhibits simple harmonic motion, the equation for the
period is
T = 2"
L
g
Theoretically, a pendulum exhibits simple harmonic motion only when the angle
(amplitude) is small and there is no drag force, so be sure that your initial angle is
! behaves as a simple harmonic oscillator.
small enough that the system
Measurement
*This has already been done in part 1.
L
(cm)
30
40
50
60*
70
80
90
100
1) We will be testing the approximate lengths indicated in the table above, be sure to
measure the actual distances as best you can. If you feel you can measure the
distances more accurately using a different technique, be sure to notify the TA
during class and note your technique in your lab report.
2) Again, the overall measurement technique is the same as in Parts 1. Be sure to
collect 5 periods for each trial and do 5 trials for each length.
Analysis
We don’t want to plot T vs L, we want to plot T2 vs L. This is because L is the
independent variable and T2 is linearly related to L, while T is related to L by a
power of 0.5.
1) The table you create will be nearly the same as in part one. The difference will be
that instead of finding the average period for each length, you will find the
average of T2. You will also find the sd and sdm of T2.
___
Plot T 2 along y and L along x. You should print your graphs. In the event that
you cannot print your graphs, write down all information the program gives
you and sketch your graph.
The slope that the program gives you should be:
4! 2
slope =
g
Given this information you should calculate g from the slope of your line. You
should also calculate the error in g from the error in the slope.
Since there is no intercept (the “b” in the equation y = mx+b) in our
theoretical equation, the value for the intercept in our experimental fit should
not differ from zero. Is it? What is the error in g?
Tips For Error Analysis and Conclusion
Error Analysis:
On average, humans have a reaction time of 0.25 seconds to visual
stimulus. (csm.jsm.edu). How will this affect your estimate of uncertainty? There
are other factors you should consider as well: how does the experiment in practice
differ from the ideal system that we wrote our equations for in class? One
example: the bob doesn’t necessarily swing in a plane; there could be motion
normal to the idealized motion of the system. Will that affect your experiment?
Try to think of other differences and attempt to quantify their affect on your
results.
Conclusion:
Did the experiment work? Does the theoretical value for g lie within one
or two SDMs of your experimental value for g? You can just restate your answers
from the analysis section to answer the above questions; you don’t have to
recalculate anything.