CP CALC A square, conducting, wire loop of side $L$ , total mass $m,$ and total resistance $R$ initially lies in the horizontal $x y$ -plane, with corners at $(x, y, z)=(0,0,0),(0, L, 0),(L, 0,0)$ and $(L, L, 0) .$ There is a uniform, upward magnetic field $\vec{\boldsymbol{B}}=\hat{B hat{\boldsymbol{k}}}$ in the space within and around the loop. The side of the loop that extends from $(0,0,0)$ to $(L, 0,0)$ is held in place on the $x$ -axis; the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a)
Find the net torque (magnitude and direction) that acts on the loop when it has rotated through an angle $\phi$ from its original orientation and is rotating downward at an angular speed $\omega$ . (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through $90^{\circ} ?$ Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain.