Oscillations and Waves > Forced and damped oscillations > Problem 3.5.42

Statement

At time , a particle of mass is subjected to a force in the -axis direction, and a force in the -axis direction. Find the trajectory of the particle if at the initial moment it is at rest. What is the average velocity of the particle over a large time? What initial velocity must the particle have in order to move in the presence of these forces on a circle? What is the radius of this circle?

Solution

From and and the initial conditions and at , we have

The last two equations are parametric equations of a cycloid (shown in blue), which is the trajectory of a point on the rim of a disk of radius that rolls without slipping on the -axis. Its average velocity is in the positive direction. If the initial velocity is in the negative direction, which is perpendicular to the initial force, then the time-dependent force will keep it in a circular orbit of radius (shown in red) as the equation for centripetal force is satisfied.

Answer

See the last paragraph in the solution.