Plotting the position of a pendulum

[bassplayer142]

Does anyone know where I could find some information about solving a pendulum position. What I mean is a swinging pendulum and the position. The position would be the spot on the ground that a sun directly above would cast. I don't really know where to start other then the pythagorean theorum, Ke and Pe equations. Any help would be great. thanks

 

[HallsofIvy]

The simple pendulum satisfies the differential equation $d^2\theta/dt^2$= -(g/l) sin(\theta)[/itex] where g is the acceleration due to gravity, l is the length of the pendulum, t is the time, and $\theta$ is the angle the pendulum makes with the vertical. Assuming the sun is directly overhead the shadow of the pendulm bob will be the horizontal coordinate, $l cos(\theta)$.

That is an extremely difficult equation to solve but for small angles, $sin(\theta)$ is approximately equal to $\theta$ so the equation can be approximated by $d^2\theta/dt^2= -(g/l)\theta$ which has general solution
$$\theta(t)= C cos(\sqrt{g/l} t)+ D sin(\sqrt{g/l} t)$$.

Determine C and D by the intitial value of $\theta$ and the angular velocity.

 

[bassplayer142]

thanks a lot!