Suppose a current loop of area a carrying current i , with moment of inertia I is placed in a uniform magnetic field of magnitude B . What is the period of small oscillations in the field.

This is essentially the problem of the simple pendulum applied to a current loop. The dipole moment p will want to align itself to an external magnetic field. The torque is:
tau=m xx B=-iaB sin(theta)=I alpha=I d^2/dt^2 theta
The small oscillations will be when sin(theta)~~theta . Then we have the differential equation:
-iaB theta=Id^2/dt^2 theta
d^2/dt^2 theta+(iaB)/I theta=0
Let omega^2=(iaB)/I
d^2/dt^2 theta+omega^2 theta=0
This has a solution of the form:
theta(t)=theta_0 cos(omega*t+phi)
Therefore the period os small oscillations is:
T=1/(f) =1/(omega/(2pi))=(2pi)/omega=2pi sqrt(I/(iaB))
https://www.acs.psu.edu/drussell/Demos/Pendulum/Pendula.html