Physics Simulation

Ball on a String

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Problem:

A ball of mass m is attached to a "non-stretchy" string of length r. It is being whirled in a vertical circle (of radius r). If the ball is going just fast enough so that the string does not go slack at the top of the circle, how fast will it be going at its lowest point?

The ultimate goal is to reach an analytic solution of the problem - to express v_bottom in terms of v_top, m, r, and whatever else (if anything) matters in the situation. A simulation can help you by:

• familiarizing you with the physical situation,
• helping you figure out what matters and what doesn't, and
• checking your final solution.

This simulation is easy to set up and run, but you may find that it is difficult for a numerical simulation to provide extreme accuracy - especially over a long time interval.

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The Simulation:

Here's a method for setting it up:

1. Open the Interactive Physics^TM program.
2. Create a circle object to represent the ball.
3. Create a rope object (), attaching one end to the center of the ball, and the other end to the background.
4. Attach a velocity meter to the ball.
5. Open the ball's Properties Window, and give it an initial x-velocity.
6. Set accuracy to "Accurate".

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Running the Simulation:

1. Run the simulation.
2. Adjust the initial velocity of the ballto find the minimum velocity that will keep the string taut. You will find that the ball will have lost considerable speed if you let the simulation run until the ball gets back to the top of the circle. This is caused by numerical and round-off errors accumulating as the simulation runs. To combat this:
   1. stop the simulation when it reaches the ball reaches the bottom of its arc.
   2. try reducing the animation step.
   3. use a more accurate integrator (Runge-Kutta 4)
3. When you have the simulation running acceptably, try systematically varying one of the quantities that you think affects the ball's speed at the bottom of the circle:
   1. mass
   2. radius
   3. g
   4. whatever...
4. Be sure to keep your data in data tables.
5. When you have collected some data, you might want to use a graphical analysis program such as Graphical Analysis^TM or DataStudio^TM to determine the relationship between each quantity and v_bottom.