Physical Measurements, Lab Experiment 5:
Determination Gravitational Acceleration with A pendulum
Experiment
Objective:
Determine the gravitational
acceleration g from the simple pendulum’s period.
Apparatus:
Meter stick, protractor, pendulum, timer.
Theory:
A simple pendulum
consists of a small mass (a bob) attached at the end of a massless string. The smallness of the bob and the massless
string assumption simplifies the mathematical solution to the problem, and with additional assumption that the
amplitude of the oscillations is also small, yields the following value for the
period (T) of the pendulum:
_{} or we can rewrite
this as
_{}
where _{} is the length of the
pendulum, and g is 9.8 m/s^{2}.
Note that pendulum’s
period squared (T^{2}) is proportional to its length l and g.
Pendulum is set up
for you.
A discussion About Precision Requirement in
Period Determination:
Since the T^{2}
is proportional to g, assuming infinite precision in length l, the percentage
error in g is twice that of in T;
_{}
This means that if
we require, say a 10% precision in determining g, the percent error in T must
be
_{}
Therefore T must be
determined with a precision of 5 %.
Since we will time the motion of the pendulum for several oscillations,
the total time t must also have 5% precision. Now, we ask the following
question. If the time determination can be done with 0.5 seconds (this should
include the instrumental uncertainty as well as our reaction time for stopping
the timing) precision, how many seconds should we time it in order to obtain a
5% precision in the period?
The answer can be
found from,
_{}
Therefore, we must
measure the time more than 10.0 seconds to accomplish our goal of measuring the
gravitational acceleration g with 10% precision. We will time approximately 10 oscillations.
Procedure:
Set the string
length to be 80 cm, measured from the fixed point to the center of the ball.
Swing the pendulum by releasing it at rest from an angle θ=20^{o}
to perform a test run. This angle is measured from the vertical direction as
shown in the figure, while the string is kept taut. Make sure that for each of the runs described below, the starting
conditions (no push, same angle) are as identical as possible. Record your data
into the table provided below by timing 10 oscillations, and repeating it for
other pendulum lengths. You can change
its length by loosening the nut at the fixed point where string attaches to the
fixed platform at the top. One oscillation is the cycle of the pendulum’s
motion starting from its initial position, and coming back to the same
position. When taking the data you
enter your measurements into the second and third columns. The other columns should be prepared by
using error propagation formulas.
Length(cm) 
Number of Osc. 
Total time t(s)±σ_{t} 
Period T(s) )±σ_{T} 
T^{2}(s^{2}) )±σ_{T}^{2} 
80 




75 




70 




65 




60 




55 




50 




45 




40 




35 




When you got this far, you are ready to make a graph of T^{2} versus l. This graph will include uncertainties as error bars. Your instructor will go over this procedure of determination of maximum and minimum slopes in class.
Your report should include the following in addition to the graph and the data table.
Best estimate from your slope g_{exp} = _______________
Expected value of period (from your textbook) g_{th} = _________
%error = _{}= ________________