A small marble is rolling on the inside of a parabolic bowl whose surface is given by z = ®r2 in cylindrical polar
coordinates. There is no friction, but the gravity is pulling down (in the negative-z direction). Consider the most
general case where the marble is free to move in any direction on the surface.
(a) Identify all the forces and write down the projection of Newton's equations on the axes of your coordinate system.
(b) Use the constraint(s) and/or integration to reduce the system of these three scalar equations to one scalar equation
in terms of r alone. (You don't have to solve this equation).
(c) Can we find the trajectory of the marble by using the energy conservation instead of integrating the r-equation
we have just obtained?
a) let the slope that the tangent to the surface at a certain point make an angle x with the z=0 plane. then tanx=d(ar2)dr=2ar
This job finds the trajectory of the marble by using the energy conservation.