TEsting PEndulum Models

This lab is designed to align with AAOT science outcome #1: Gather, comprehend, and communicate scientific and technical information in order to explore ideas, models, and solutions and generate further questions.

Materials

• string or fishing line
• dense object with compact shape (nut or washer, rock, toy car)
• sturdy structure for the object to freely swing from (cabinet handle, horizontal railing, shower curtain)
• protractor (here is one you can print)
• writing utensil
• calculator
• digital device with spreadsheet program
• digital device with internet access

ObjectiveS

1. Understand the major limitations of the simple pendulum model caused by the assumptions contained in the model.
2. Apply the simple pendulum model to experimental data from a variety of real pendula made from objects with different shapes and densities.
3. Treat the gravitational acceleration near Earth’s surface (g) as a free parameter of the simple pendulum model and extract a value.
4. Compare the model prediction for g  to the accepted value.
5. Evaluate the overall success of the simple pendulum model in predicting the frequency of real pendula under specific conditions.
6. Analyze the time dependence of the pendulum amplitude using a damped oscillator model.

The simple Pendulum Model

The simple pendulum model predicts that the pendulum will behave as a simple harmonic oscillator with the angle of the string away from vertical serving as the displacement. The oscillator frequency will depend only on the length of the pendulum (L) and the gravitational acceleration (g) according to the following equation, which we will put to the test in this lab.

$ f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$

The simple pendulum model is derived under the following assumptions:

1. The swinging object is a point mass.
2. The string (or rod) used to hang the object has no mass.
3. There is no air resistance or friction in the system.
4. The amplitude is small enough for the small angle approximation to apply ($sin\Theta \approx \Theta$).

1) Determine the angle in radians above which the small angle approximation has more than 1% error. (At what angle does the percent difference between $sin\Theta$ and $\Theta$ become more than 1%).

2) According to the simple pendulum model, the the oscillation frequency depends on the string length to what power? (Length is raised to what power in the frequency equation at the start of the lab?) Note that no dependence on length means a power of zero. Explain.

3) According to the simple pendulum model, the the oscillation frequency depends on the mass to what power? Explain.

4) According to the simple pendulum model, the the oscillation frequency depends on the oscillation amplitude to what power? (Meaning amplitude is raised to what power in the frequency equation?)  Explain.

Amplitude Experiment

We will test the simple pendulum model by verifying that it correctly predicts how the oscillation frequency on length, mass, and amplitude.

5) As you found above, the model is only valid for amplitudes less than _____radians or, equivalently, ______degrees. For the purpose of this lab we will call that angle $\Theta_1$. Hang your pendulum and release it from an amplitude less than $\Theta_1$  and use a stopwatch to time 10 full oscillations. Record your data below.

Object:__________

Starting amplitude:__________

Time for 10 Oscillations:__________

6)  Use your time for 10 oscillations to calculate the oscillation period,  frequency, and angular frequency. Show your work.

Amplitude Data

7) Repeat the experiment for every amplitude from 2 degrees up to $\Theta_1$ at 2 degree intervals. Record your results in a spreadsheet and use the spreadsheet to efficiently calculate the period and frequency for each case.

Amplitude Analysis

8) Make a graph of the oscillation frequency vs. amplitude. Be sure to label the graph and the axes, including units.

9) Apply a best fit line and record the fit function and R² value here. Does the frequency appear to depend on amplitude? Explain.

Amplitude Conclusions

10) With regard to the dependence of oscillation frequency on amplitude, does the data support the validity of the frequency equation predicted by the simple pendulum model? Explain.

Length Experiment

Length Data

11) Shorten the string by about 10% and choose an amplitude less than $\Theta_1$. Repeat the procedure find the oscillation frequency by timing 10 oscillations at least 6 more times, shortening the string by a bit each time.  Record the string length and oscillation frequency in your spreadsheet. (Do not repeat the amplitude test for each string length). Be sure to measure pendulum length from the rotation point all the way to the center of mass of the object, not just the string length!

Length Analysis

12) Make a graph of oscillation frequency vs. length. Make sure that your length data has units of meters. Be sure to label the graph and the axes, including units. Copy and paste your frequency and length data into two new columns of the online spreadsheet.

13) Apply a power law fit to data in the graph of  frequency vs. length. Record the fit equation and
R² value here:

14) Does the power law model provide a good fit to your data? Explain.

15) According to your fit equation, what is the power on the length variable? Does this agree with the power dependence predicted by the model? [Hint: You plotted frequency vs. length, so “y” in your fit equation is actually “f” and “x” in your equation is actually “L”. Compare the power of x in the fit equation to the power of L in the frequency equation for a simple pendulum]

16) Compare your fit equation to the simple pendulum equation. Rewriting your fit equation, but replacing “y” with “f” and “x” with “L”  will help you remember what the variables in the fit equation mean. If you then write the simple pendulum frequency equation directly below that, you can directly compare each part of each equation. The fit equation indicates that frequency is equal to a coefficient (just a number) multiplied by the length raised to some power. The simple pendulum equation predicts that the frequency is equal to a combination of constants multiplied by the length raised to a power (-1/2). Therefore, if the fit is good, then the combination of constant in the pendulum equation should be equal to the coefficient in the fit equation. Set these equal and solve for g, and calculate a numerical value.

17)  Applying a simple pendulum model to your data has allowed you to extract a value for the free-fall acceleration due to gravity on the surface of Earth without having to actually drop anything in free-fall. Of course your pendulum wasn’t really a perfect simple pendulum, and there is uncertainty in your measurements, both of which lead to error. Compare the value for g that you found with the accepted value for g. Show your work in calculating a % difference.

Length Conclusions

18) With regard to the dependence of oscillation frequency on length, does the data support the validity of the the simple pendulum model? Explain.

MAss Experiment

Mass Data

To provide experimental data on a variety of masses we will combine all of the class data. Your chosen objects will have a variety of masses. If the oscillation frequency is independent of mass and amplitude, as predicted by the simple pendulum model, then we should be able to fit all of the frequency vs. length data with a single equation.

19) Copy and paste the data from your classmates into the frequency and length columns of your own sheet to create one combined dataset. Make a new graph of this data. Be sure to label the graph and the axes, including units.

20) Apply a power fit to the class data. Record the fit equation and R² here:

MAss Conclusions

21) With respect to mass, does the class data support the validity of the simple pendulum model? Explain.

Overall Conclusions

22) Overall, do you conclude that the simple pendulum model is valid under the conditions tested by you and your classmates? Explain in terms of the results of each set of analyses you performed.

Further Questions

23) What changes to the experiment would you make to ensure that the experimental conditions more closely matched the assumptions of the simple pendulum model? Explain the reasoning behind your proposed changes.

Modeling a Damped Pendulum

24) The simple pendulum model predicts that the pendulum will behave as a simple harmonic oscillator, which means the pendulum motion can be described by cosine or sine functions and that the amplitude is constant. However, you observed that the amplitude was not constant–the amplitude decreased over time. Explain what caused the decrease.

In some cases the damping force is proportional to the velocity of the oscillator:

$F_d = -bv$

In that special case, the amplitude will decay exponentially and the decay constant will depend on the damping constant, $b$, that defines the strength of the damping force:

$A(t) = A_0e^{-\frac{b}{2}t}$

Let’s find out if  the amplitude of a pendulum can be modeled using an exponential decay, in which case the damping forces acting on the pendulum have the form $F_d = -bv$. The following video shows a pendulum swinging.

25) Label two new with columns with amplitude and time, including units. Watch the video and record the starting amplitude for time zero in your spreadsheet. Next, record the stopwatch time and amplitude on every 5th oscillation until about 1 minute. After one minute the amplitude changes slowly enough that can begin recording the time and amplitude on every 10th oscillation. For each data point you will likely need to use the pause, scrub, and playback speed features of the video player to narrow down when the amplitude fist appeared to drop by a full 0.5 degrees. The playback speed can be adjusted by clicking on the gear in the lower right of the player. 

26) Plot the amplitude vs time, name and label the graph, and apply an exponential fit. Record the equation and R² value here:

27) Do the data suggest that the damping forces acting on the pendulum are proportional to the velocity? Explain.