Pendulum Motion - Maple Help

Pendulum Motion

Main Concept

This demonstration shows how the length of a pendulum's arm and the acceleration due to gravity influence the speed of a pendulum's motion. You can choose any value between 5 and 30 meters for the length of the arm and any value between 1 and 20 meters per second squared for the acceleration due to gravity.

 Derivation  The differential equation for the angle as a function of time $t$ is   .   It is not possible to write a formula for the solution to this equation in terms of an elementary function. Instead, we use an approximation which is fairly accurate if the angle $\mathrm{\theta }$ is sufficiently small (i.e. when $\mathrm{sin}\left(\mathrm{θ}\right)\approx \mathrm{θ}$ ):     .   Solving this differential equation allows us to find formula for the angle of the pendulum at a given time $t$ :     and the angular speed at a given time $t$ :       where ${\mathrm{\theta }}_{0}$ is the initial angle of the pendulum. Note that the angular frequency of the pendulum is a constant${\mathrm{ω}}_{0}=\sqrt{\frac{g}{L}}$.

 Length of the pendulum's arm: $\mathrm{m}$ Acceleration due to gravity:

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