Magnetism and Matter

KEY CONCEPT : The time period of a rectangular magnet oscillating in earth's magnetic field is given by $T=2\pi\sqrt{\frac{I}{\mu B_H}}$ where $I=$ Moment of inertia of the rectangular magnet$\mu=$ Magnetic moment$B_H=$ Horizontal component of the earth's magnetic field Case $1: T=2\pi\sqrt{\frac{I}{\mu B_H}}$ where $I=\frac{1}{12} M \ell^2$ Case 2 : Magnet is cut into two identical pieces such that each piece has half the original length. Then $T'=2\pi\sqrt{\frac{I'}{\mu' B_H}}$ where $I'=\frac{1}{12} \left(\frac{M}{2}\right) \left(\frac{\ell}{2}\right)^2 = \frac{I}{8}$ and $\mu' = \frac{\mu}{2}$$\therefore \frac{T'}{T} = \sqrt{\frac{I'}{\mu'} \times \frac{\mu}{I}}$$= \sqrt{\frac{\frac{I}{8}}{\frac{\mu}{2}} \times \frac{\mu}{I}}$$= \sqrt{\frac{1}{4}} = \frac{1}{2}$