Section A
Q1.
(a) Suppose the position of a particle of mass m on the xy-plane is \barr = (x, y, z) = (r cos \phi, r sin \phi, 0). Here r and \phi are functions of time t. Find lz, the z-component of \vecl, where \vecl is the angular momentum of the particle. Further, show that the areal velocity is equal to the angular momentum divided by 2m. Is areal velocity constant?
(b) Explain the rotation of a free rigid body and show that a rigid body rotating in any manner about a fixed point has two constants of motion, L2 and T. Here \vecL is angular momentum and T is kinetic energy.
(c) Two spaceships approach each other, each moving with the same speed as measured by a stationary observer on the Earth. Their relative speed is 0.7c. Determine the velocity of each spaceship as measured by the stationary observer on the Earth.
