A reference frame rotates with constant angular velocity relative to an inertial frame that has the same origin as . A particle of mass at position vector is subject to a force . Derive the equation of motion for the particle in .
A marble moves on a smooth plane which is inclined at an angle to the horizontal. The whole plane rotates at constant angular speed about a vertical axis through a point fixed in the plane. Coordinates are defined with respect to axes fixed in the plane: horizontal and up the line of greatest slope in the plane. Ensuring that you account for the normal reaction force, show that the motion of the marble obeys
By considering the marble's kinetic energy as measured on the plane in the rotating frame, or otherwise, find a constant of the motion.
[You may assume that the marble never leaves the plane.]