Lab 04 - The Simple Pendulum

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Experiment 04 – The Simple Pendulum 1
INTRODUCTION
The motivation for this lab was to examine the factors affecting the motion of a
pendulum. An ideal pendulum behaves according to simple harmonic principles as displayed in
Equation 1 given that the string is also weightless. Equation 1 notes that the period of a
pendulum depends on the length of the string, and gravity. Because gravity is assumed to be
constant here on earth, the only variable (according to Equation 1) affecting the period of a
pendulum is the length of the string. The following two investigations mathematically prove that
the period of a pendulum does not depend on the mass attached (Investigation I) and that the
period depends on the length of the string (Investigation II). The first investigation takes two
bobs with differing masses attached to a string whose length was constant, attached to a fulcrum
and measures each one’s period, timed with a stopwatch. If the assumption above is correct then
the period should not depend on the pendulum’s mass. The second investigation varies the
string’s length while holding the mass of the bob constant with similar procedures as that of
investigation I. If the mathematical principle of pendulum motion is once again correct, then the
period should vary with a varying string length. Investigation II also derives the gravitational
constant through using simple algebra which supports Equation 1.
EQUATIONS
PROCEDURE
Aspherical bob’s mass was measured with a scale and diameter with a calliper. Attached
to a string whose connection was to a rod with a string clamp mounted to a stage, the bob was
tied with an approximate length of 90cm, measured from the fulcrum to the centre of the bob.
The bob was then moved to approximately 10o away from its equilibrium position (aligned with
the stage at rest) and released. A stopwatch timed each period the bob completed for a total of
ten oscillations. This process was repeated four times to obtain 10 measures of time all noted in
the Appendix section below the Questions section. While the lab procedure instructed to measure
the total time of ten oscillations, measuring each oscillation individually was done in the interest
of brevity. Ultimate calculations and derivations should not have been affected by this deviation
in the procedure as the average time per oscillation would have had to be calculated, nonetheless.
This procedure was replicated for another bob whose mass was different from the first. All
measurements and derivations are present in Table 1.
Table 1 - Displays gathered data from the procedure described. Where all measurements
are listed in SI units. 𝛿m was calculated by noting half of the next significant figure position
unobtainable with the scale. 𝛿d was derived in the same way as 𝛿m but with the calliper. L1
refers to the length of the string whose 𝛿L was derived in the same fashion as 𝛿m and 𝛿d but
relative to the meter stick. AVG t was calculated using the average of 10-time measurements,
referenced in the Appendix. Because the AVG t was measured for each oscillation AVG t is
equal to the value of the period T. 𝛿t was calculated using Equation 6, where the root means
squared (of all the 10 data points) is divided by N, which was 10. 𝛿t is therefore also equal to 𝛿T.
T2 represents the period squared which is equivalent to AVG t squared. 𝛿T2 was calculated using
Equation 4.
Graph 1 -. Represents the relationship between the mass of the bob case studied in the
investigation and its period. Whose slope was 0.0231x + 1.89s. The points represent both masses
and the error bars represent 𝛿m.
CONCLUSION
The properties of simple harmonic motion, involving pendulum motion, was the core of
this lab. Investigation I mathematically verified that the period of the pendulum did not depend
on mass. The relationship between mass and period was determined to be 0.0231x +
1.89s±0.21s, which was to be expected as the relationship between mass and period should
theoretically be zero. The gravitational constant was derived to be 10.0m/s2±0.0942m/s2 which
approximated the theoretical 9.8m/s2 . This is to say that there was a direct relationship between
period and length evident in the slope 3.94x (±0.371) + 8.49x10-3s. One of the only factors that
could have affected the results is the value of air resistance on the bob during the pendulum
motion. It was concluded that accounting for the drag force would further approach the
theoretical values of this experiment. Improvements to this lab include accounting for drag or
using more of a variety of pendulum lengths.

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pendulum. An ideal pendulum behaves according to simple harmonic principles as displayed in

constant through using simple algebra which supports Equation 1.

measurements and derivations are present in Table 1.

and the error bars represent 𝛿m.

theoretically be zero. The gravitational constant was derived to be 10.0m/s2±0.0942m/s2 which

using more of a variety of pendulum lengths.

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