Pendulum Snake
The motion of a pendulum is not easy to describe analytically, but some simplifications (small-angle approximation) allow us to obtain simple formulas to describe its motion, period and frequency.
If we release the bob with a starting angle radian and no angular velocity, we can describe the motion of the pendulum at any instant with the formula , where is the length of the (massless) cord and the gravitational acceleration constant.
The motion of a pendulum is then harmonic, and the angle is the amplitude of the oscillation, that is the maximum angle between the cord of the pendulum and the vertical equilibrium position of the bob.
For the period of a pendulum can be approximated by the Huygen's law .
Quite counter-intuitively, the period doesn’t depend either on the mass of the pendulum bob or on how far the pendulum swings, but it depends only on the length of the cord! (see Galileo's isochronism for details).
The frequency is defined as the inverse of the period, therefore we have .
Solving this equation for , we can find which length of the cord generates a certain number of oscillations around the equilibrium position:
Try It Yourself...
You now have all the necessary "ingredients" to create cool patterns using the pendulums in the app below.
We have 6 of them, and you can use the sliders to adjust the starting angle and the length of the cords of each pendulum.
Can you figure out how many oscillations will make each bob in 30 seconds, using the formulas above and the data in the app?
Use the Play button without modifying the default lengths and angle to animate the pendulums and explore the current pattern, then use the sliders (and formulas!) to create your own pendulums.