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
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 sinθ≈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 :
ⅆθⅆ t=θ0⋅ gLsingL⁢t
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:
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