[UPSC IFoS 2011] Conservation Laws (Q34, 10M)

A particle of mass 'm' moves according to the equations x = a \cos \omega t, y = a \sin \omega t, z = c, where a, c, \omega are constants. Obtain the instantaneous velocity and linear momentum vectors in terms of the Cartesian components and hence the angular momentum. Find the force \vec{F} and the torque \vec{N} acting on the particle and verify that the angular momentum \vec{L} and the torque \vec{N} satisfy the relation \frac{d\vec{L}}{dt} = \vec{N}.