Superconducting Loop

Disregarding any sign conventions:

FLux through the loop at any instant:

$\phi = a^2 B_z$

Therefore magnitude of induced EMF:

$\lvert \dot{\phi} \rvert= a^2 \alpha \dot{z}$

Using Kirchoff's voltage law:

$a^2 \alpha \dot{z} = L\dot{I}$

When $t=0$ , $z=0$ and $I=0$ . Integrating:

$a^2 \alpha z = LI$

At any instant of time, the total magnetic force along the vertical ( $F=IaB_x$ applied for side of the loop and added):

$F_B = -\alpha a^2 I$

Newton's second law:

$m\ddot{z} = F_B -mg$

Simplifying: $\ddot{z} + \frac{\alpha^2 a^4z}{mL} = -g$

$z(0) = \dot{z}(0)=0$

General solution:

$z = A \cos\left(\frac{\alpha a^2t}{\sqrt{mL}}\right) + B \sin\left(\frac{\alpha a^2t}{\sqrt{mL}}\right) -\frac{mLg}{\alpha^2 a^4}$

Applying initial conditions and solving for $A$ and $B$ :

$\boxed{z = \frac{mLg}{\alpha^2 a^4}\left(-1 + \cos\left(\frac{\alpha a^2t}{\sqrt{mL}}\right)\right)}$