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.

 

Derivation

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:

m

Acceleration due to gravity:

m s2

 

  

 

 

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