Question #43186

A small cart of mass $m$ is constrained to move along the positive x-axis ($x > 0$) on a frictionless horizontal track. It is attached to the origin by a light spring of force constant $k = 6.4 \text{ N/m}$ (natural length is zero). Mounted on the cart is a small concave mirror of focal length $f = 40 \text{ cm}$. The mirror is mounted on a frictionless pivot, allowing it to rotate freely in the xy-plane. A magnetic dipole of moment $\mu = 2.0 \text{ A m}^2$ is rigidly attached to the mirror. The dipole vector $\vec{\mu}$ points along the principal axis of the mirror, and the reflecting surface of the mirror faces the direction $-\vec{\mu}$. The system is placed in a non-uniform magnetic field $\vec{B} = (\alpha x)\hat{i} + B_1\hat{j}$, where $\alpha = 4.0 \text{ T/m}$ and $B_1 = 3.0 \text{ T}$. A point object $O$ is permanently fixed at coordinates $(-0.6 \text{ m}, 0.8 \text{ m})$ in the xy-plane. When the cart and mirror reach their stable equilibrium position and orientation, what are the coordinates of the image of $O$ formed by the mirror? (Assume the mirror is small enough that the magnetic field is uniform over its volume).