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Lab 11: Damped Oscillations
T.A:
Monday at 2:00 p.m
Abstract: This experiment was done in order to determine the effects amplitude, length and
damping have on the motion of a pendulum. For part one, a tennis ball was attached to varying
lengths of string and the period of motion for ten oscillations was found. For part two, the period
of ten oscillations for varying amplitudes of the tennis ball pendulum were found. For part three,
the motion of a Styrofoam ball pendulum was recorded for two minutes. From part one, an
experimental value for the acceleration of gravity was found to be 12.5 +/- .1 m/s^2. Although
that value is much larger than anticipated, the error was largely due to human error in the
recording with the stopwatch. For part two, by graphing the amplitude versus the period there
was no correlation found as the slope was close to zero (.003 +/- .002 sec/deg). For part three the
values of τ, w, Q, and f were approximated by looking at the graph and the estimates were
compared to the values given by the linear fit ( .8 versus .73 Hz for frequency, 5.026 versus 4.58
rad/sec for w, 40 versus 43 seconds for τ and 201.04 versus 199.55 radians for Q). Since the
results were very similar, it was determined that both methods were acceptable but the method
relying on the linear fit was suggested as the error was much less for the linear fit estimates.
Overall the objectives of this lab were met despite human error with recording of the period. Objective:
Examine how period varies with pendulum length and amplitude and to analyze how the motion
is affected by damping.
Equipment:
The equipment used in this experiment was a tennis ball and a Styrofoam ball set up as a
pendulum with varying lengths for the string. For parts one and two, a stopwatch was used to
record the period and Data Analysis was used to record the results. A sonar detector was used
for part three to record the period over two minutes.
Procedure:
First, the diameter and mass of the tennis ball and Styrofoam ball were recorded. For part 1,
the period of ten oscillations were found for varying lengths of string for the tennis ball using
amplitude less than .1 radians. For part 2, the period of ten oscillations for five different
amplitudes, between five and twenty degrees, for the tennis ball was found. For part 3, two
minutes of motion with the Styrofoam ball of a length of near sixty centimeters and amplitude of
less than eleven degrees was recorded using a sonar detector.
Results:
Part 2 Theoretical Value for Gravity Actual value
12.5 +/- .1 m/s^2 9.81 m/s^2
Eyeball Actual Value
Approximation
f .8 Hz .73 Hz
ω 5.026 rad/sec 4.58 rad/sec
40 seconds 43.57
τ
seconds
Q 201.04 rad 199.55 rad
Data Analysis:
Part 1: T=2∗π L T =4∗π ∗L/G T vs.L=¿ π 2 4∗π ¿/G
√ G (4* *L/G) *1/L slope = (
2
4∗π
G= slope
Part 3:
Approximated f by counting the number of oscillations in the first 10 seconds.
Approximated τ by finding the time it took for the graph to decay to e^(-1) times its maximum
−1
value. The maximum amplitude was .14 so to find τ, e ∗.14 was found to be roughly .5 and
the x-coordinate corresponding to that value was roughly 40 seconds.
A e−x/sin(cx+d)
The oscillatory fit equation found from the graph was
The value obtained for c is equal to the value of , so ω= 4.58 rad/sec.
The value of b is equivalent to τ, so τ=43.57 sec.

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