A magnetic dipole of moment M and moment of inertia I is initially held at 120° to a uniform magnetic field B and released from rest. Neglect friction.
Its maximum angular speed is:

Question

A magnetic dipole of moment M and moment of inertia I is initially held at 120° to a uniform magnetic field B and released from rest. Neglect friction. Its maximum angular speed is:

G1 Physics · Accepted Answer

The dipole has maximum angular speed while passing through the stable equilibrium position θ=0. Magnetic potential energy: U = -MB cosθ. Initially, U i U_i U i ​ = -MB cos120° = +MB/2. At stable equilibrium, U f U_f U f ​ = -MB. The decrease in potential energy is U i U_i U i ​ - U f U_f U f ​ = MB/2 + MB = 3MB/2. This becomes rotational kinetic energy: ½Iω_max² = 3MB/2. Therefore, Iω_max² = 3MB, so ω_max = √(3MB/I).