A bar magnet is oscillating in the Earth's magnetic field with a period T. What happens to this period and motion if its mass is quadrupled -

For small oscillations of a magnet in a uniform field, the restoring torque is proportional to the angular displacement, so the motion is simple harmonic with
$T = 2\pi\sqrt{\dfrac{I}{MB}}$, where I is the moment of inertia and M the magnetic moment.
The moment of inertia is proportional to the mass ($I \propto$ mass, dimensions unchanged), so when the mass is quadrupled, $I' = 4I$. The magnetic moment is unchanged.
$T' = 2\pi\sqrt{\dfrac{4I}{MB}} = 2 \times 2\pi\sqrt{\dfrac{I}{MB}} = 2T$
So the motion remains simple harmonic, with time period 2T.