Physics Exercises/Rotational Kinematics

1. 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.)

2. 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.