A compass needle is joined to an axle which permits it to turn freely in the horizontal plane so that merely the horizontal component of the magnetic field influences its motion. The magnetic moment of the needle is around 9.0x10^-3 A.m^2, its moment of inertia is 2.0x10^-8 kg.m^2, and the horizontal component of the Earth's magnetic field is around 1.9x10^-5 T.
(i) Determine the torque on the needle as a function of angle θ among the needle and the north direction?
(ii) Determine the frequency of small oscillations of the needle around the north direction?