Question #43292

A magnetic needle of moment of inertia $I$ and magnetic dipole moment $M$ is suspended horizontally by a uniform wire of length $L$ and radius $r$ in a uniform horizontal magnetic field $B$. The needle undergoes small angular oscillations in the horizontal plane. Initially, the torsional restoring torque per unit radian of the wire is exactly equal to the magnetic restoring torque per unit radian. The temperature of the entire system is then increased by a small amount $\Delta \theta$. The coefficient of linear expansion of the needle is $\alpha_1$, and that of the suspension wire is $\alpha_2$. The shear modulus of the wire decreases with temperature as $\eta = \eta_0(1 - \gamma \Delta \theta)$, and the magnetic moment of the needle decreases as $M = M_0(1 - \delta \Delta \theta)$. Assuming $\Delta \theta$ is small enough that higher-order terms are negligible, what is the fractional change in the time period of oscillation, $\frac{\Delta T_p}{T_p}$?