Final answer:
The one-dimensional problem equivalent to the motion is simple harmonic motion (SHM). The condition for circular motion is that the gravitational force equals the centripetal force.
Step-by-step explanation:
The one-dimensional problem equivalent to the motion of a particle of mass m constrained to move under gravity without friction on the inside of a paraboloid of revolution whose axis is vertical can be represented as a simple harmonic motion (SHM). In SHM, the displacement of the particle from equilibrium is proportional to the restoring force acting on it. In this case, the force of gravity provides the restoring force.
To produce circular motion, the particle's initial velocity should be such that it provides the necessary centripetal force to counteract gravity. The condition for circular motion is that the gravitational force equals the centripetal force:
mg = mv^2/r
where v is the initial velocity and r is the radius of the circular path. Rearranging the equation, we can solve for the initial velocity:
v = sqrt(g * r)
The period of small oscillations about this circular motion can be found using the formula:
T = 2 * pi * sqrt(r/g)