PhysicsLAB: A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus

Equipment

Each lab group will need the following equipment:

two vertical support stands with
one horizontal cross bar
one slotted rod
a triple beam balance
a meter stick
a LabPro
a motion detector

diagram courtesy of Daniel Weaver (c/o 2008)

Background Information

This experiment is a simple exploration of the physical pendulum. We will start our investigation by using a “long rod” as our model. During the experiment, you will be using a metal rod with pre-drilled holes. Recall that the moment of inertia of a long "thin" rod about its center of mass is

If we were being precise in our physical model the equation for a rectangular-type rod would be more appropriate. The moment of inertia for this more exact physical model is

where a represents the length of the rod and b is the rod's thickness.

Refer to the following information for the next three questions.

Physical Data: For your group's aluminum rod, measure a and b in cm and the rod's mass in grams.

length = a = ____ cm

thickness = b = ____ cm

mass = ____ grams

Refer to the following information for the next three questions.

Comparing Moments: Calculate the rod's moment of inertia with each formula and compare your results.

What is the rod's moment of inertia using

What is the percent difference between these two values for your rod's moment of inertia through its center of mass?

I suspect that you will agree that in most cases where b << a, our first approximation, , will work just fine. From this point forward we use this simpler formula.

Data Collection

The experiment is fairly simple. You need to measure the period of your physical pendulum for each pivot point above its center of mass, cm. I would recommend measuring either 5 or 10 full oscillations for small amplitudes less than 10º.

As you complete your trials, fill out the following chart. Compute each period and measure each "Y" and "D" value to three decimal places.

Now we will use EXCEL to plot T² vs D by opening 1-physicalpendulum.xls. Remember to save your group's copy as LastnameLastnameLastnamePhysicalPendulum.xls in your period's folder.

What is the filename of your EXCEL spreadsheet?

Print your EXCEL graph and locate the minimum of the graph on your plot and verify its coordinates with your data chart. Highlight the closest values on your chart and then type the coordinates of the minimum position on your EXCEL graph in the blank provided.

Analysis and Conclusions

The classic expression for the period of a physical pendulum is

For our apparatus the distance above the pivot point was called "Y" and the distance from the pivot point to the center of mass was called "D." Therefore we can express the formula for the period of our physical pendulum as

where the parallel axis theorem helps us calculate

In our experiment we used a constantly changing axis of rotation which meant that the moment of inertia of our rod was also constantly changing.

Refer to the following information for the next question.

Use the parallel axis theorem,

, the moment of inertia of a thin rod about its center of mass, and the period of a physical pendulum given above to prove that the period of a physical pendulum that is pivoted a distance “D” away from its center of mass is given by:

Have your teacher initial and date your submission when this step has been completed on your papers.

Refer to the following information for the next three questions.

Now square both sides of the expression you derived above to get an expression for T², the y-axis value on your EXCEL graph.

Using calculus, determine the theoretical value of D that would produce the smallest value of T² for your physical pendulum. Remember that L, g, and pi are constants. Show your work on your EXCEL graph.

D = ____ meters

Using the value for D that you discovered in the previous question, evaluate the theoretical minimum value of T for your pendulum. Show your work on your EXCEL graph.

T = ____ seconds

Calculate the percent error between this theoretical value for the pendulum's minimum period and your graphical experimental value. Show your work on your EXCEL graph.

% error =

Each group's lab report should include a cover page, a copy of your data table (with its highlighted row), and a well-annotated copy of your EXCEL graph showing all required calculations.

This lab is used with the permission of its designer:
     David Jones
     Miami Palmetto High School
     Seminole Community AP Physics Institute
     July, 2005