Note: When running this simulation on a tablet or a smart-phone, set the parameters and enable the Accelerometer, then press the Play button. When running this simulation on a computer, set the parameters and disable the Accelerometer, then press the Play button and drag the slider. You may experience a slight delay when moving the slider.

Ball on a Moving Ring

The Ball on a Moving Ring mobile JavaScript model illustrates the forces and dynamics of a ball rolling to a circular track. This simulation uses the accelerometer on your mobile device to read the proper acceleration as measure by the device. The proper acceleration is the acceleration relative to a freely falling observer whereas coordinate acceleration is dependent on the reference frame acceleration and this acceleration (a_x, a_y, a_z) is displayed in the simulation. Note that the proper acceleration for a stationary object on Earth is approximately +9.8 m/s^2 because the object is accelerating up as seen by a freely falling observer. The mobile device accelerometer measures proper acceleration because the device is acted on by gravity and by the device acceleration. Subtract the acceleration due to gravity to obtain the coordinate acceleration as defined in introductory physics textbooks. 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 mobile device 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, disable the Sensor to simulate back and forth motion with the slider. 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 force from the track (blue), the gravitational force (black), and the force of friction (green) can also be displayed. You can set the initial angle in radians and the angular velocity in radians/second and you can adjust the viscous damping (friction) coefficient. Test your knowledge of circular 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 rolling on a track. The mobile device acceleration or the slider when the Sensor is disabled will cause the ball to move. The ax, ay, and az values are the proper acceleration of the track reference frame. There is also a low speed viscous drag (friction) force F_drag=-Dv that is proportional to the speed v and the damping (friction) coefficient D. Setting a non-zero friction coefficient is useful to damp out transient motion when making observations.

Assume the y-axis of the mobile device is vertical. 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 if the y-axis is vertical? How does this value change if the device is tipped around the x-axis?
Tilt the device back and forth. Does this motion produce an 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 ball to roll 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 the 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.