The period of oscillation of a freely suspended bar magnet...

The period of oscillation of a freely suspended bar magnet is $4 s$. If it is cut into two equal parts in length, then the time period of each part will be
[a) $4 \mathrm{~s}$
(b) $2 \mathrm{~s}$
(c) $0.5 \mathrm{~s}$
(d) $0.25 \mathrm{~s}$

Key Concepts

Simple Harmonic Motion of a Magnetic Dipole

A bar magnet suspended in a magnetic field experiences a restoring torque when it is displaced from equilibrium, causing it to oscillate. For small angular displacements, these oscillations behave like a simple harmonic oscillator, where the time period is determined by the balance between the magnetic restoring torque and the magnet’s moment of inertia.

Moment of Inertia

The moment of inertia quantifies an object’s resistance to angular acceleration, depending on both its mass and the distribution of that mass relative to the pivot point. When a uniformly distributed bar magnet is split into two equal pieces, each piece has a smaller mass and a shorter length, leading to a reduced moment of inertia compared to the original magnet.

Magnetic Dipole Moment

The magnetic dipole moment is a measure of the strength and orientation of a magnet's magnetic field, and it is typically proportional to its size or length when magnetization is uniform. Cutting the bar magnet in half reduces its magnetic dipole moment, which directly affects the restoring torque experienced in the magnetic field.

Dependence of Oscillation Period on System Parameters

The period of oscillation for a magnetic dipole is given by the formula T = 2??(I/(mB)), where I is the moment of inertia, m is the magnetic moment, and B is the magnetic field. Changes in either I or m, such as through physically altering the magnet's dimensions, lead to different oscillation periods.

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