Note: When running this simulation on a tablet or smart-phone, set the parameters, then press the Play button. When running this simulation on a computer, be sure to uncheck “Accelerometer,” then set the parameters and press the Play button.

Pendulum with Moving Support

The Pendulum with Moving Support mobile JavaScript model illustrates the forces and dynamics of a pendulum attached to a moving support. This simulation uses the accelerometer on your mobile device to read the proper acceleration as measure by the mobile device. The proper acceleration is the acceleration relative to a freely falling observer whereas coordinate acceleration is dependent on the choice of reference frame acceleration. Note that the proper acceleration for an object at rest relative to Earth is +9.8 m/s^2 in the up direction as seen by a freely falling observer. The mobile device accelerometer measures proper acceleration because it is acted on by gravity and by the device acceleration but it is interesting to note that proper acceleration is fundamental in Einstein's general theory of relativity because this theory assumes that gravity is equivalent to acceleration in spacetime.

You may need to lock the orientation of your screen to maintain a fixed view as you tilt and (gently) accelerate your device. For use on computers without an accelerometer or for classroom use, the "No sensor" mode simulates back and forth motion. Search for "mobile JavaScript model" in the Open Source Physics digital library to find other simulations that use mobile device sensors.

A force body (free body) diagram showing the pendulum rod force (blue), the gravitational force (black), and the force of friction (green) can also be shown. Users set the initial angle in radians and angular velocity in radians/second and you can adjust the viscous damping (friction) coefficient when the simulation is paused. Test your knowledge of pendulum motion by answering the following questions and comparing your answers with the simulation results. Remember to press the play button after you set the initial values.

Initial values, such as θ₀ and ω₀, can be set when the simulation is paused but not when the simulation is running. (Non-editable input fields have a grey background.) You may also drag the mass when the simulation is paused. The reset button returns the system to its default state while the reset time button returns the system to the previous values of θ₀ and ω₀.

Physics Exercises:

This simulation shows a mass m=1 handing from a rod of length L hanging from a moveable support point. The support point's motion is either the motion of the mobile device or the motion of the slider when the "no sensor" option is selected. There is also a low speed viscous drag (friction) force F_drag=-Dv that is proportional to the speed v and the damping coefficient D. Setting the non-zero drag is useful to damp out transient motion when making observations.

Draw force body diagrams labeling all forces for the pendulum mass at θ = { 0°, 45°, 90° 180° } without friction and with ω=0. Note that the angular position is entered using radians, not degrees, in the simulation.
How will your answer to 1 change if ω is not zero?
What is the small angle frequency of oscillation in radians/sec and in cycles/sec?
Tilt the device back and forth. Does this motion produce a pendulum oscillation? Explain.
Push and pull your device back and forth. Does this motion produce a pendulum oscillation? Explain.
How often (what frequency) should you push and pull your device to cause the pendulum to swing over the top? Does it make a difference when you push and when you pull?
Enter a larger initial angular velocity and observe the motion and free body diagram. Describe the push-pull motion needed to stop the pendulum motion.

Note: Continuously reading the accelerometer may drain the mobile device battery. Uncheck "Accelerometer" or navigate to a different HTML page to conserve battery power. The sensor should stop when navigating to another browser page but this behavior is platform dependent.

Teaching and Learning Objectives:

Write the force law for pendulum motion involving physical properties such as the mass, the rod length, and the acceleration due to Earth's gravitational field.
Solve this equation for small angle oscillations using either sines or cosines.
Describe the effect of a moving support.
Suppose you are holding your mobile device while riding on a merry-go-round. How can you determine the angular velocity of the merry-go-round using measurements from the provided by the simulation if you are sitting a distance R from the center of the merry-go-round.