$($a$)$

A $12\mathrm{.}0$ m long, thin, uniform aluminum pole slides south at a speed of $28\mathrm{.}0$ m$/$s$.$ The length of the pole maintains an east$-$west orientation while sliding. The vertical component of the Earth's magnetic field at this location has a magnitude of $50\mathrm{.}0$T$.$ What is the magnitude of the induced emf between the ends of the pole $($in mV$)?$

mV

$($b$)$

What If$?$ The west end of the pole impacts and sticks to a pylon, causing the pole to rotate clockwise as viewed from above. While the pole rotates, what is the magnitude of the induced emf between the ends of the pole $($in mV$)?($Hint: use conservation of angular momentum to find the speed of the pole after the collision.$)$

Given the formula for the emf from part $($a$),$ can you construct a formula for the emf of an infinitesimal slice of the pole in terms of radial distance from the pivot and angular speed? Integrating this, what is the emf in terms of angular speed and length? What is the initial angular momentum of the pole $($based on its center of mass speed$)$ about the pivot point? What is the final angular momentum as it rotates $($based on the moment of inertia of a pole rotating at one end$)?$ Use this to find the final angular speed, and therefore final emf. mV