Question #87795

A small carriage of mass $M$ can move along a frictionless horizontal track aligned with the $x$-axis. It is connected to the origin ($x=0$) by a horizontal spring of spring constant $K$, which is relaxed when the carriage is at $x=0$. A small magnet with magnetic dipole moment $\mu$ and moment of inertia $I$ (about the vertical $z$-axis) is suspended from the carriage by a vertical torsion fiber of torsional constant $C$. The magnet can rotate freely in the horizontal $xy$-plane. The entire system is placed in a static, non-uniform magnetic field given by $\vec{B} = (B_0 + \beta x)\hat{i} - \beta y\hat{j}$, where $B_0$ and $\beta$ are positive constants. Initially, the system is in equilibrium with the dipole moment aligned along the positive $x$-axis. If the magnet is given a small angular displacement in the $xy$-plane, what is the angular frequency $\omega$ of the resulting torsional oscillations?