Lab Activity: Develop a Math Model for the Period of a Pendulum

Purpose:  In this activity you will investigate how hanging mass, length, the strength of gravity and starting angle affect the period of a pendulum. Then, form a math model for the period of a pendulum.

Materials:         
Pendulum apparatus              Stop watch or other timing device/technique

Meter stick                           Various masses for pendulum bob

Balance                                 Protractor or other means of measuring angle
Logger Pro or Tracker          Electronic force sensor         Video camera or phone

Background:

                A pendulum is a classic example of periodic motion, which is a motion that is repetitive, redundant, keeps repeating itself, keeps going back and forth…oh, I’ll stop now.  A pendulum only works when there is gravity, and of course can be used as a clock.  It is also an example of circular motion, which means there is a net centripetal force, mv²/R.  Because we began studying circular motion and will be getting deeper into gravity, we will try to figure out the basic properties of a pendulum and see how we can combine gravity and circular motion together in a few different ways, whether it is a pendulum, an amusement park ride, or satellite motion.  Keep the force diagram for a pendulum in mind (what does it look like?).

Research Questions:

What effect(s), if any, do mass of the bob, length of the pendulum, and angular amplitude of a pendulum, have on the period of the pendulum?  Our goal is to use mathematical fits to data to determine an empirical formula for the period of a pendulum.  Your group is looking to combine your graphical fits into the form T = f(L)g(angle)h(m)s(g).

Develop Hypothesis:

Before doing any measurements, state what you think the effect of mass, angular amplitude, length, and the acceleration of gravity have on the period of the pendulum.  Remember, reality always tends to be a bit complex!

Procedure:

                You will be trying to determine the best fits to data using Excel.  Your measurements will use four different techniques for collection. 

i.                     Use a stop watch to measure the period as a function of length of the pendulum;

ii.                   Use video to determine the period as a function of mass hanging from the pendulum. NOTE: when changing masses, you will need to adjust the length of the string accordingly to maintain the same total length to the center of mass;

iii.                  Use an electronic force sensor to measure the period as a function of angular amplitude;

iv.                 Use the PhET simulation for pendulums to determine the effect of gravity on the period – check out the blog post which provides the links and information for this portion.

 You will need to get data to make four graphs:

i.                     Period as function of length. Note the constant mass and angle;

ii.                   Period as function of mass. Note the constant length and angle;

iii.                  Period as function of starting angle. Note the constant mass and length;

iv.                 Period as function of g (using PhET simulation, keep constant mass and length and angle).

Keep in mind, you should always be thinking of estimating errors on all measurements (in any experiment you ever do!) as well as how to minimize those errors.  Think about what might be the best way of measuring a given quantity.  Also, do not forget to include units on all measurements, along with a reasonable estimate of the uncertainty of all measurements!  For anything measured in trials, think standard deviation.

                For your write-up: Purpose, Materials, Procedure, Data Tables, Questions and Analysis (with appropriate graphs; these should be titled and labeled with quantities being graphed and units!  Graphs need to be done on the computer.).  Do a group report, and consider Google Docs if that works best for the group.

Definitions:

                Period = time it takes the pendulum, when released from rest, to swing over and back to where it started; or the time for one round-trip.

                Length = length from the point where the string is free to swing to the center of mass of the bob at the end of the string.

                Mass: we assume massless string, so just the mass of the hanging bob.

                Angular amplitude: the starting, maximum angle for the period, as measured from the vertical.

Questions/Analysis:

1.       For the graph of period as a function of length, figure out errors on measurements and include them as error bars on your graph.  The error bar for period measurements (some number of trials with the stop watch) should be the standard deviation, as always.  Find the best-fit function for the graph in Excel. Include the equation and R² value.  Include the point (0.01,0.01) on your graph and in your fit, since (0,0) won’t allow certain fit options.

2.       For the graph of period vs. mass, use five different masses and use a video to get period measurements in Logger Pro or Tracker.  Make sure to adjust the length of string so the overall lengths of the pendulum are always the same.  For instance, when you hang a larger mass on the string, you will need to shorten the string a bit since the object will be longer (think about where the center of mass is).  Try to estimate errors in length and in period from your video measurements in LoggerPro or Tracker. Find a best-fit function for the data.  Include the equation and R² value.

3.       For the graph of period vs. angle, use the electronic force sensor and measure the period in the Logger Pro display. Plot your data using Excel and obtain a best-fit function to the data to get an idea of the relationship. Go from small angles around 10^o, and go up to around 90^o.

4.       Last but not least, graph period vs. g from the PhET simulation.  Find a best-fit function for the data.  Include the equation and R² value.  Keep in mind you will only have 3 points, so do what you can with this.

5.       There are only three significant forces acting on the mass at any time, those being tension, gravity and air friction.  Which of these are constant, which are non-constant?  If any are non-constant, when are they the strongest, and when are they the weakest?  Explain, and include a force diagram/free-body diagram for a pendulum.

6.       Using your graph in #1, how long should a pendulum be in order to have a period of 1.0 sec (on earth)?  How long should a pendulum be to have a period of 2 seconds?  Use your fit to do so.  Keep in mind the first mechanical clocks were made with the pendulum! (way to go Galileo)

7.       What exactly does F_centripetal = mv²/R mean?  What does the value of mv²/R tell us, and what does

             it depend on specifically for a pendulum?  Think about what force or forces keep the pendulum

             moving in that portion of a circle.  Consult our notes and the book for assistance.

Write a brief, to-the-point (i.e. one paragraph) summary of your findings/conclusions about what the period of the pendulum depends on and how it depends on specific quantities. 

*Use and combine the results of your individual best-fit functions for your four graphs to obtain an empirical mathematical model for the period of your pendulum as functions of length, mass, g, and angular amplitude.*

              Put it in the form: T = f(L)g(angle)h(m)s(g)

Assess the techniques.  Based on your experience:

-          Rank the ease of the four measuring techniques.

-          Rank the accuracy of the four measuring techniques.

Provide any other feedback you think is relevant to this set of experiments!