A particle of mass

*m*is constrained to move on a horizontal circle of radius*R*. This circle also rotates in space about a fixed point (*P*) on the circumference of the circle. The rotation of the circle is about an axis of rotation perpendicular to the plane of the circle and tangent to the circle, at point (*P*); the rotation is at a constant angular speed*ω*.Show that the particle’s motion about one end of a diameter passing through the pivot point and the center of the circle, perpendicular to the axis of rotation, is the same as that of a plane pendulum in a uniform gravitational field.

(

*Note: This is a difficult question, most likely requiring the use of Lagrangian mechanics, not simple Newtonian mechanics.*)- What is the average density of a planet if an orbital period of a spacecraft on a low orbit is
*N*minutes? Assume that by "low orbit" means that the spacecraft's orbital radius approximately equal to the planet's.