Pendulum Motion - Maple Help
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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.



The differential equation for the angle as a function of time t is


ⅆ2θⅆ t2+gLsinθ=0.


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 θ is sufficiently small (i.e. when sinθθ ):


  ⅆ2θⅆ t2+gLθ=0.


Solving this differential equation allows us to find formula for the angle of the pendulum at a given time t :


θt=θ0 cosgLt


and the angular speed at a given time t :


ⅆθⅆ t=θ0 gLsingLt 


where θ0 is the initial angle of the pendulum. Note that the angular frequency of the pendulum is a constantω0=gL.



Length of the pendulum's arm:


Acceleration due to gravity:

m s2





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