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A Rigid Threebar Pendulum
Developed by E. Behringer
This set of exercises guides the student in exploring computationally the behavior of a physical pendulum consisting of three bars. It also requires the student to generate, observe, and describe the results of simulating the rotational motion for different configurations of the pendulum. The numerical approach used is the halfstep approximation (a modified Euler) method. Please note that this set of computational exercises can be affordably coupled to simple classroom experiments with meter sticks.
Subject Area  Mechanics 

Level  Beyond the First Year 
Available Implementation  Python 
Learning Objectives 
Students who complete this set of exercises will be able to
* express an equation predicting the period of small oscillations in terms of dimensionless ("scaled") variables suitable for coding (**Exercise 1**);
* produce both contour plots and 1D plots of the period of small oscillations versus scaled variables (**Exercises 1 and 2**);
* derive the equation of motion for the pendulum (**Exercise 3**);
* computationally model the motion of a threebar pendulum with damping using the halfstep approximation integration algorithm (**Exercise 4**);
* produce graphs of the computational solution, i.e., of the angular position versus time, for different geometrical configurations of the pendulum (**Exercise 4**); and
* assess the accuracy of the computational solution by comparing it to the analytical solution for small oscillations and no damping (**Exercise 5**).

Time to Complete  120 min 
### Exercise 1: Dependence of the period of small oscillations on scaled characteristics of the bars
A rigid, threebar pendulum can be made by fastening together a single bar of mass $m_1$ and length $l_1$ to two identical bars of mass $m_2$ and length $l_2$ as shown below.
![](images/threebar_pendulum/Threebar_pendulum_Fig1.png "Optional Title")`
![](images/threebar_pendulum/Threebar_pendulum_Fig2.png "Optional Title")
For the undamped, rigid threebar pendulum, it is possible to show that the period of small oscillations is given by
$$
T=2\pi \left[ \frac{ \frac{1}{3} m_1 l_1^2+2\left[\frac{1}{12} m_2 l_2^2+m_2 \left(l_1s+l_2/2\right)^2 \right]} { (m_1+2m_2)g \frac{m_1 l_1/2+2m_2 (l_1s+l_2/2)}{m_1+2m_2} } \right]^{1/2}
$$
Show that this is equivalent to the expression
$$
T=2\pi \left[ \frac{l_1}{g} \frac{ \frac{1}{3}+2\tilde{m}_2 \left[\frac{1}{12}\tilde{l}_2^2+\left(1\tilde{s}+\tilde{l}_2/2\right)^2 \right]} {\frac{1}{2}+2\tilde{m}_2 (1\tilde{s}+\tilde{l}_2/2)} \right]^{1/2}
$$
where the "scaled variables" $\tilde{m}_2$, $\tilde{l}_2$, and $\tilde{s}$ are defined as $\tilde{m}_2 \equiv m_2/m_1$, $\tilde{l}_2 \equiv l_2/l_1$, and $\tilde{s} \equiv s/l_1$. Assume $l_1=1.0$ m and $\tilde{l}_2=1.0$ to generate a contour plot of the period $T$ versus $\tilde{m}_2$ on the vertical axis and $\tilde{s}$ on the horizontal axis, with $0.5 \leq \tilde{m}_2 \leq 2.0$ and $0.0 \leq \tilde{s} \leq 1.0$.
Does the value of the period change as you would expect as $\tilde{m}_2$ and $\tilde{s}$ are increased?
### Exercise 2: Dependence of the period of small oscillations when the bars are identical
If the bars comprising the threebar pendulum are all identical ($\tilde{m}_2 = 1$), show that the result of Exercise 1 simplifies to
$$
T=2\pi\left[ \frac{l_1}{g} \frac{1+(32\tilde{s})^2}{1+2(32\tilde{s})} \right]^{1/2}
$$
Plot the period $T$ versus the scaled overlap $\tilde{s}$ for $0 \leq \tilde{s} \leq 1$ when $l_1=1.0$ m. What is the value of the period when $\tilde{s}=0$ (no overlap)? When $\tilde{s}=1$ (total overlap)? Would it be difficult to experimentally determine the difference between these values with a stopwatch?
### Exercise 3: Equation of motion of the damped threebar pendulum
If the damping of the pendulum is modeled as a resistive torque represented by $Rv_{cm}v_{cm}$, where $v_{cm}$ is the velocity of the center of mass of the pendulum and $R$ is a constant, show that the equation of motion of the damped pendulum is
$$
\ddot{\theta} = \frac{mgl_{cm}}{I} \sin\theta  \frac{Rl_{cm}^2}{I} \left\dot{\theta}\right \dot{\theta}
$$
where $\theta$ is the angular position of the center of mass of the physical pendulum measured counterclockwise from the downward vertical direction, $m$ is the total mass of the bars, $g$ is the gravitational acceleration, $l_{cm}$ is the distance from the rotation axis to the center of mass of the physical pendulum, $I$ is the rotational inertia of the physical pendulum with respect to the rotation axis, $R$ is the proportionality constant for the damping, and dots represent differentiation with respect to time. Furthermore, show that, if we define the dimensionless time $\tilde t \equiv t/T$ where
$$
T=2\pi\left(\frac{I}{mgl_{cm}}\right)^{1/2}
$$
then the equation of motion assumes the dimensionless form,
$$\frac{d^2 \theta}{d\tilde{t}^2} = 4\pi^2\sin\theta 
\frac{R l_{cm}^2}{I} \left \frac{d\theta}{d\tilde{t}} \right \frac{d\theta}{d\tilde{t}}$$
### Exercise 4: Solve the equation of motion of the damped threebar pendulum
(a) Solve the equation of motion from Exercise 3 using the halfstep approximation algorithm if the pendulum is released from rest at $t = 0$ s. Plot the resulting solution in the case that the bars are all identical, have a scaled overlap of $\tilde{s}=0.5$, experience no damping ($R = 0$), and the pendulum is released from rest at an angle of $\theta_0 = 45^\circ$ measured from the vertical.
(b) Repeat part (a), but for a scaled overlap of $\tilde{s}=0.25$.
(c) Repeat part (a), but with nonzero damping given by $R=0.1$ kg. Check that the oscillations diminish in amplitude with time (or else something is wrong with the code!).
### Exercise 5: Accuracy of 1D Model Angular Position vs. Time
Because the computational solution is based on an approximation, it is important to determine just how small $\Delta t$ should be for the approximation to accurately predict the angular position. You can do this by comparing the computational solution to the exact solution when the oscillations are small and damping is negligible. In this case, the equation of motion reduces to $\ddot{\theta}=\frac{mgl_{cm}}{I}\theta$. If the pendulum is released from rest from $\theta = \theta_0$ at $t = 0$ s, this equation has the analytical solution
$$
\theta(t)=\theta_0 \cos(2\pi t/T)
$$
where the period $T$ is given by
$$
T=2\pi\left(\frac{I}{mgl_{cm}}\right)^{1/2}
$$
Compare the time dependence of the angular position predicted by the computational model to that predicted by the analytical solution for small oscillations when the undamped pendulum consisting of three identical bars is released from rest at an angle of $\theta_0 = 5^\circ$ measured from the vertical, and the scaled overlap is $\tilde{s}=0.5$.What value of $\Delta t$ do you deem to be sufficiently small for the computational model to be accurate? Explain how you arrived at this value of $\Delta t$ .
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Credits and Licensing
E. Behringer, "A Rigid Threebar Pendulum," Published in the PICUP Collection, July 2016.
The instructor materials are ©2016 E. Behringer.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license