The physics
Spin a rod about one end, with the field along the spin axis. It makes an emf. Each bit moves at v = ω r and adds B v dr. Add them from the pivot to the tip.
ε = ½ B ω ℓ²
A rod pivoted between its ends has two arms. Both push charge the same way from the pivot. So the ends differ by the difference of the arms.
Vends = ½ B ω |a² − b²|
A disc is many rods side by side, from the hub to the rim. So is a wheel of spokes. Side by side, they give one rod's emf, like cells in parallel.
A pendulum wire turns about its hinge. It turns fastest at the bottom. Energy gives that turning speed.
ωbottom = √(2 g (1 − cos θ₀) / L)