b
(b)\(T = 2\pi \sqrt {\frac{I}{{M{B_H}}}} = 4\,sec \)
When magnet is cut into two equal halves, then New magnetic moment \(M' = \frac{M}{2}\)
New moment of inertia \(I' = \frac{{\left( {{\rm{w}}/2} \right){{\left( {l/2} \right)}^2}}}{{12}} = \frac{1}{8}.\frac{{{\rm{w}}{l^2}}}{{12}}\)
Where \(w\) is the initial mass of the magnet
But \(I = \frac{{{\rm{w}}{l^2}}}{{12}};\;\;\therefore \;I' = \frac{I}{8}\)
\(\therefore \) New time period \(T' = 2\pi \sqrt {\frac{{I'}}{{M'{B_H}}}} \)
\( = 2\pi \sqrt {\frac{{I/8}}{{\left( {M/2} \right){B_H}}}} = \frac{1}{2}2\pi \sqrt {\frac{I}{{{M_H}}}} \)\( = \frac{1}{2} \times T = \frac{1}{2} \times 4 = 2\,sec \)