4. Velocity-Dependent Forces II: Sliding in a Bowl with Friction

The motion of a particle sliding a hemispherical in a bowl falls somewhere between one- and two-dimensional problems. The motion is along the arc of a circle in two dimensions (if the particle is released from rest), but only one variable is needed to specify position, and, if there is no friction, only the tangential force enters the calculation. Without friction the problem seems simple, but it (the large amplitude pendulum) is a very challenging analytic problem. When friction is added, no analytic solution is possible - the numeric solution is the only solution.

We must use the Feynman algorithm extension developed in the last section to deal with this problem because the radial component of the force provides the centripetal acceleration, and this makes the frictional force in the tangential equation velocity-dependent. The writing of the expression for the tangential acceleration requires some care, but there is nothing complicated about the rest of the program. (A fine time interval is used to avoid worry about what, if anything, should be corrected in the intervals where the velocity reverses.)

The Java program plots displacement versus time, but the QuickBasic program simply tables the turning points and the stopping point. The program needs only a minor change to introduce different values for the coefficients of static and kinetic friction.