Large Amplitude Period of a Physical Pendulum
Ben Kinnaman, Davidson College
Jeff Weeks, Davidson College
February 9, 1998
For small amplitude oscillations, the period of a simple pendulum may be easily expressed as 2p Ö (l/g). This holds true however, only when the amplitude remains very small. When the amplitude becomes large the period of the pendulum becomes amplitude dependent. In this experiment we used a rotary motion sensor interfaced with a software package to record, analyze, and compare with theory the large amplitude behavior of a pendulum. The data we recorded closely matches the theoretical results and proves that as the amplitude of a pendulum increases, so must its period.
Though for this experiment we used a physical pendulum, for the discussion of the theory behind the experiment we will consider a simple pendulum to simplify the mathematics initially. After we have gone through the derivation of the theoretical large amplitude period, we will address the issue of the rigid body problem.
First, consider a particle of mass m attached to a weightless, rigid rod of length l that is constrained to rotate vertically about a point in a gravitational field g (Figure A).
Since the gravitational field acts downward on the particle, there is a force exerted on the particle. The force resulting in the motion of the particle acts perpendicular to the rod as a component of the gravitational force. This force is F(q ) = -mgSinq , noting that we have declared the force of gravity to be acting in the negative direction. There is also a torque acting on the rod. The torque is equal to the tangential component of force multiplied by the moment arm, or t = lF. We can also define the torque as the product of the moment of inertia, I, and the angular acceleration, a : t = Ia . The following derivation gives us an equation of motion for the pendulum.
F(q ) = -mgSinq
Ia = lF = -mglSinq
The moment of inertia, I, is defined as I = ml2
ml2 a = -mglSinq
a= - w 02Sinq or a + w 02Sinq = 0
where w 02 is defined as g/l
Now, we have an equation for the motion of the pendulum, a + w 02Sinq = 0. For small amplitudes we may approximate Sinq = q . Thus, the equation of motion is just as it is for the simple harmonic oscillator, a + w 02q = 0. Using this expression the period is given just as it is for the simple harmonic oscillator, t = Ö (l/g).
We have made a very limiting assumption by approximating the period for small amplitudes only. What about for any finite amplitude that allows oscillatory motion, be it linear or non-linear? The equation for the period that we have derived thus far is independent of amplitude, provided that the amplitude remains small enough for our small angle approximation to hold true. The exact solution for the pendulum motion shows, however, that as the amplitude increases so will the period. We will now remove the requirements that the amplitude remain small and discuss the exact motion for a pendulum. Note that in the following derivations the position at which the pendulum is at rest is 0, and for now, the initial position will be defined as q 0.
To solve themotion of the pendulum, we will begin with the expressions of the Kinetic, T, and Potential, U, energies.
The kinetic energy of the particle is
T = 1/2 Iw 2 angular expression, T = 1/2 ml2 q '2
The Potntial energy expression is derived from the torque
E = constant
T + U = E = constant
1/2 ml2 q '2 - mglCos(q ) = E
When the pendulum is at its maximum amplitude of an oscillation or, q 0, the angular velocity is zero and thus, the Kinetic energy is zero, the energy expression becomes the following at the pendulums amplitude
E = - mglCos(q )
Since the angular velocity, q '2, is dq /dt, we can make the following rearrangements substituting the energy at q 0 into our conservation of energy expression
We now need to transform the above into an elliptical integral in order to obtain a solution for the pendulum motion with unrestricted amplitude. The following substitution is made using the identity: Cosq = 1- 2*Sin2 q /2.
Substituting this into our integral we get
We need to make some rearrangements to prepare to solve for the period
If we let the pendulum start at the lowest point of oscillation, where q = 0, and go until the amplitude, q , we have one fourth of the period. The expression then becomes
Let's make the following substitutions,
Substituting into the last expression of the period, t , we have a very manageable integral
Using this expression we will be able to substitute in our experimental amplitudes as q 0. This will give us our theoretical periods as dependent on the amplitude.
Below is a schematic of the complete apparatus used for this experiment.
A.This is the Pasco rotary sensor and physical pendulum used for the experiment. The sensor is attached to the axle of the pendulum. We used the sensor to record angular position as a function of time. The entire device was securely clamped to the table edge. The pendulum used was a rigid pendulum with a weight attached to one edge.
B.This is the interface unit made by Pasco Scientific to interface the rotary motion sensor and the Scientific Workshop software package. Channel 1 and Channel 2 were both used as specified by the instructions for both the equipment and the software.
C.The interface unit was connected to the PC as specified. The PC ran the Science Workshop software package by Pasco. The software allowed us to record the angular position as a function of time and to plot the data as well. All data was directly recorded using this software package/interface combination. Data was further analyzed in an Excel spreadsheet. No system requirements for the PC are relevant to the experiment outside of what the software specifies.
Because the bearings of the pendulum axle are in sealed races with lubrication inside, it was necessary to give the pendulum a few good spins in order to lubricate the bearings evenly before taking any measurements. This insured a greater uniformity in the data. Taking data consisted starting the pendulum swinging with an initial amplitude, q 0. While the pendulum was in motion the Scientific Workshop software and the Pasco Rotary Motion Sensor recorded the angular position at a rate of 200 samples per second. Data was then transferred from the Scientific Workshop desktop to a spreadsheet for analysis. Each of these steps is explained below in greater detail. Note: for the purpose of our discussion, the position at which the pendulum is at rest will be referred to as 0 radians.
The pendulum was initially started at a very small initial amplitude, q 0, in order to determine the small angle period of the pendulum. Data was subsequently recorded for 20 initial amplitudes. All initial amplitudes were approximately evenly spaced over a range starting at 0.07 radians and ending 0.37 radians shy of p radians. Since we were starting the pendulum swinging by holding it with our fingers it was imperative to begin the data sampling only when the pendulum was being held steady at an absolute reference point.
The data is displayed in the Scientific Workshop on a graph showing angular position as a function of time and in a table. In the table the sampled angular position was listed in chronological order and displayed showing 6 significant digits. All data was extracted from the table to insure the greatest possible accuracy. We recorded the initial amplitude and the amplitude of the first swing. We also recorded the period of the first oscillation and the time it took to complete half of the first oscillation.
In the spreadsheet the amplitude was calculated by the average of the initial amplitude, q 0, and the amplitude of the first swing of the pendulum, q 1. The error is shown on the plot of the data as the difference between the average and q 0 (positive) and the difference between the average and q 1 (negative). The period, t , corresponding to each q 0 is the period for one oscillation. The error for the period was figured as the difference in the time it took to complete each half of the oscillation. The data was then displayed in a plot of period as a function of amplitude and compared to the corresponding plot of the theoretical period as a function of amplitude. To calculate the theoretical period we used the expression
But, we made several adjustments to this expression. First, we had to address the problem of us using a physical pendulum, a rotating rigid body, and our calculations being based on a simple pendulum, a particle in a circular path. For very small amplitudes the period of oscillation is given by the small angle approximation.
For a physical pendulum, this expression is different but what is important is that the small angle approximation holds true. Thus, we were able to substitute our experimental period for a very small amplitude as the coefficient in front of the integral. We run into a problem though, our experimental small angle period really has a 2p term included. To compensate for that, we were able to numerically integrate the following expression
where kj are the experimental amplitudes substituted into the expression for k derived in the theory section. (Please see the theory section for a complete discussion of this expression.) Note that 1.085 was the period found for very small amplitudes. The MathCad analysis of this expression is available.
Below is the data collected during this experiment and the data upon which all conclusions and data analysis is made. The first column, labeled "Data Amp" is the initial amplitude with which we gave the period. The amplitudes are actually averages of the initial and first oscillation amplitudes, as discussed above. The second column, labeled "Data period" is the recorded period of the first oscillation. The third column, labeled "Series Expansion" is the theoretical set of periods derived by solving the following expression for the period as a function of amplitude:
As is evident this expansion did not closely match our data for larger amplitudes. We instead numerically integrated the following expression for the period as mentioned before.
The theoretical periods resulting from this evaluation are shown in the fourth column, marked "Numerical Analysis". It is obvious that this method closely fits our data.
The plot of this data is shown below. The fuchsia markers designate the theoretical values for the period while the blue markers represent our experimental data. The error bars shown on experimental data points shown the deviation of the average amplitude from the amplitudes of the initial and first swings. The error bars representing the error in the period represent the difference between the time required to complete the first and second half of the first oscillation. It is not important to display the calculations of the errors, it is important to know that the errors were due to frictional dampening of the pendulum occurring from non-ideal mechanical behavior in the rotary motion sensor and the pendulum itself.
The data that we collected for the period of this pendulum very closely match what the theory suggests through numerical analysis. Text books state the amplitude dependence of the period in terms of the first couple of terms in the above expansion, but we found that accurate only up to amplitudes of about p /4. While some error is present at the larger amplitudes, we feel that this is due to perhaps an imprecise measurement of the small angle period, further human error in measurements, and non-ideal mechanical behavior. By numerical analysis we were able to confidently determine the large amplitude dependence of the period and prove that as the amplitude of the pendulum increases so does the period, as suggested by the theory.
Arya, Atam P. Introduction to Classical Mechanics, Second Ed., New Jersey: Prentice Hall. 1998.
Marion, Jerry B. and Stephen T. Thornton. Classical Dynamics of Particles & Systems, Third Edition. Orlando: Harcort Brace Jovanovich. 1988.