Pendulum with Varying Length. A pendulum is formed by a mass m attached to the end of a wire that is attached to the ceiling. Assume that the length $I(t)$ of the wire varies with time in some predetermined fashion. If $\theta(t)$ is the angle in radians between the pendulum and the vertical, then the motion of the pendulum is governed for small angles by the initial value problem

$$\begin{array}{l}{l^{2}(t) \theta^{\prime \prime}(t)+2 l(t) l^{\prime}(t) \theta^{\prime}(t)+g l(t) \sin (\theta(t))=0} \\ {\theta(0)=\theta_{0}, \quad \theta^{\prime}(0)=\theta_{1}}\end{array}$$

where g is the acceleration due to gravity. Assume that

$$l(t)=I_{0}+l_{1} \cos (\omega t-\phi)$$

where $l_{1}$ is much smaller than $l_{0}$ l0. (This might be a model

for a person on a swing, where the $pumping$ action

changes the distance from the center of mass of the swing to the point where the swing is attached.) To simplify the computations, take $g=1$Using the Runge-Kutta algorithm with $h=0.1$ study the motion of the pendulum when $\theta_{0}=0.05, \theta_{1}=0, \quad l_{0}=1, l_{1}=0.1$ $\omega=1,$ and $\phi=0.02 .$ In particular, does the pendulum

ever attain an angle greater in absolute value than the

initial angle $\theta_{0} ?$

## Discussion

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## Recommended Questions

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$$

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$$

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\begin{array}{l}

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\end{array}

$$

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