Question #43287

A thin rectangular plate of mass $2m$, length $L$ (spanning $0 \le x \le L$), width $2a$ (spanning $-a \le y \le a$), and uniform thickness $t$ has a non-uniform magnetization given by $\vec{M} = M_0 \left( \frac{x}{L} \hat{i} + \frac{y}{a} \hat{j} \right)$. The plate is cut into two pieces along the diagonal line connecting $(0, -a)$ and $(L, a)$. The upper fragment (the piece containing the point $(0, a)$) is taken and reshaped into a uniform solid sphere of mass $m$ and radius $R$. During this reshaping, its total magnetic dipole moment vector $\vec{\mu}$ is preserved and becomes permanently fixed relative to the sphere's material. The sphere is placed in free space and is initially at rest. A short magnetic field pulse $\vec{B}(t) = B_p \delta(t) \hat{k}$ is applied, where $\delta(t)$ is the Dirac delta function. Find the magnitude of the angular velocity of the sphere immediately after the pulse.