# Pendulum Modeling Activities

Install Ejs and then right-click within the program to copy the Free Fall model from this package into the Ejs workspace to do the following activities.

### Period

Set the initial angle to 30 degrees (π/6 radians). Is the period longer or shorter than what is predicted for simple harmonic motion? What is the percent difference between the pendulum period and simple harmonic motion? Repeat for 90 degrees (π/2 radians) and 180 degrees (π radians).

### Angular Velocity

Predict the shape of the angular displacement graph for angles if the bob is placed very near the vertical. Run the simulation to test the correctness of your prediction. Predict the shape of a dθ/dt plot for this same initial condition. Add an a plot dθ/dt to the simulation and compare its shape with your prediction.

Note that both ω= (g/L)1/2 and dθ/dt have units or radians per second and both are sometimes referred to as angular velocity. These quantities are, however, quite different. Omega ω is a constant that determines the rate of change of the phase the simple harmonic oscillator analytic solution whereas is dθ/dt is the instantaneous rate of change of the displacement angle and is not constant

### Phase Space (Advanced)

A phase space plot shows angular position θ vs. angular momentum L = I dθ/dt as the pendulum evolves. Add this plot to the simulation. Explain the shape of the plot for small angles. Explain the shape for large angles. What happens if the pendulum is given sufficient angular momentum (angular velocity dθ/dt ) so that it passes over the top?