Comparison of Exact and Approximate Solutions to the Pendulum

G1: Exact and Approximate Pendulum Equations

The exact equation of motion for the pendulum, as we discussed in the introduction, is

The term sin(Theta) in this equation makes it hard to solve analytically. For small angles,

, and the approximate pendulum equation of motion

is much easier to solve.

Before starting the analysis, start the pendulum (set to "exact") at several angles, including as straight up as you can manage, and see qualitatively how the period and frequency of the motion changes. How large can you make the period? How small?

(G1a) The Frequency for the Linear Approximation

(G1a.1) Calculate the frequency in cycles per second for the approximate pendulum equation of motion, as a function of the mass, g, and L. Add the formula to your writeup. Evaluate it for the parameters of our problem (Mass=50 gm, g=9.81 m/s^2, and L=10 m).

(G1a.2) Change the menu on the upper left hand corner of galileo from "exact" to "approx". The pendulum program is now simulating the approximate pendulum equation of motion. Start at Theta = 0.7, ThetaDot = 0, (and of course leave the damping alpha=0), and run. Measure the period for one oscillation, to at least three significant figures.

Zoom in on the graph with the right mouse button, to get a good measure of the period. (At large magnifications, you can see the line segments that the computer uses in its numerical solution to the equation.) To restore the original size, click with the right mouse button. Hold the left mouse button down at a point to get the coordinates.
You will find it easier to get the period from the plot of ThetaDot vs. Time. Measuring zero-crossings is easier than measuring the location of a local maximum.
If you want a more accurate answer, try changing the time step to a smaller value. Select "Configure", and type in 0.01 for dt, and then type Return. This will slow the program down: for those of you with fast machines, that might be a good thing. Change it back to 0.05 afterward, if you like.

Add the period to your writeup.

(G1a.3) From the period, compute the frequency given by the simulation. Compare to the frequency given by the analytic formula, and include this comparison in your writeup. Make sure they agree to the accuracy you expect from the simulation! Does the frequency change as you increase the initial angle? If you pick an initial angle greater than pi, you'll see a qualitative difference between the exact and approximate solutions!

(G1b) Comparing the Frequencies: Exact vs. Approximation

(G1b.1) Change the menu on the upper left hand corner of galileo from "approx" to "exact". Galileo is now simulating the exact pendulum equation of motion. Again, starting at Theta = 0.7, ThetaDot = 0, measure the period for one oscillation, to at least three significant figures. The motion should look quite close to a sinusoidal motion, but with a slightly different frequency from that of the approximate solution. Write the period in your writeup. Use the "Copy Graph" button to open a new window and store the graph there. Using the "configure" button on the new window, label the curve "Exact" and change the color. (You should probably use "black", "gray", or "light gray". "Turquoise" will work, but might print out the same color as "green" on a gray-scale laserprinter.)

(G1b.2) Change back to "approx", and run from the same initial conditions. Using the "steal data" button on the new graph window, load the approx curve into the stored graph. If you already haven't done so, close the configure window for the graph, and then use the "configure" button again to label the second curve. Does your name appear on the graph as the title? Use the "save postscript..." button to store this graph, and make a printout for your writeup unless your TA tells you otherwise. (You'll need a PostScript printer, or a translator which converts for your printer. CIT has PostScript printers available for a fee per page.)

(G1b.3) Is the exact period larger or smaller than the approximate one? Is the exact frequency larger or smaller than the approximate one? In your writeup, explain this difference. Do so in four ways:

By expanding sin(theta) to third order in theta, explain how the difference between the approximate and exact values for ThetaDotDot should lead to the observed frequency difference.
By clicking with the left mouse button, start the pendulum at a value of Theta nearly straight up. (What value of Theta is this?) How does the period of the approximate motion change? Of the exact motion? What would happen if you started straight up? In this extreme case, does the direction of the frequency shift agree with that seen at small values of the initial Theta?
Plot (as above) the potential energy vs. theta, for both the approximate and exact equation of motion, on the same graph. (The range will be determined by the initial value for Theta: pick it nearly straight up.) Label the curves, print the graph, and include in your writeup. Which potential energy grows more quickly? What should that do to the frequency?
Plot the torque vs. theta, for the exact and approximate equation of motion, and include in your writeup. A larger torque should produce a more rapid response. Does that help explain the frequency shift?

(G1c) Pendulum Frequency vs. Amplitude

We know that the solution to the approximate equation of motion is a sinusoid, where the frequency of the sinusoid is independent of the maximum amplitude of oscillation. However, you might expect that the frequency shift that you observed in the previous section is a function of the maximum amplitude of the swing of the pendulum. This is because in the limit as the initial Theta becomes very small, the approximation becomes better and better.

Set the simulation to "exact". Generate the solution to the exact pendulum equation of motion for the initial value Theta_0 equal to 0.05, 0.3, 0.9, 1.2, and 1.8, and measure their periods, using the methods noted above. (You need not make a combined plot for all these curves.) Using your own program (or graph paper!), make a plot of frequency vs. initial amplitude and include in your writeup.

The main message of this part of this problem is that the harmonic approximation sin(theta) ~ theta is an approximation! It works remarkably well for small displacements, but for larger amplitudes it breaks down. For the rest of this course, we will be studying wave equations based on this same linear, harmonic approximation. For most systems, the wave equation has important corrections if the waves have large amplitudes --- just like the pendulum. For very large amplitudes, the differences between the linear theory and the nonlinear theory can become qualitative, not just quantitative.

(G1d) Qualitative Change

Start with an initial value of Theta of three, and an initial value for ThetaDot equal to one. What does the motion look like now? In your writeup, explain how this motion is qualitatively different from the oscillations around the bottom of the well seen for the rest of this problem.

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