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/s2.

Note that pendulum’s period squared (T2) is proportional to its length l and g.

Pendulum is set up for you.

A discussion About Precision Requirement in Period Determination:

Since the T2 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=20o 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)±σt Period  T(s) )±σT T2(s2) )±σT2 80 75 70 65 60 55 50 45 40 35

When you got this far, you are ready to make a graph of T2 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 gexp = _______________

Expected value of period (from your textbook)  gth = _________

%error = = ________________