The time period of a freely suspended magnet is a 4 seconds....

The time period of a freely suspended magnet is a
4 seconds. If it is broken in length into two equal parts and one part is suspended in the same way, then its time period will be
(a) $4 \mathrm{sec}$
(b) $2 \mathrm{sec}$
(c) $0.5 \mathrm{sec}$
(d) $0.25 \mathrm{sec}$

Key Concepts

Magnetic Pendulum Oscillations

This concept involves the oscillatory motion of a suspended magnet in the presence of a magnetic field. The restoring torque on the magnet arises from the interaction between its magnetic moment and the external magnetic field. The dynamics are analogous to those of a pendulum, and the time period is generally expressed by an equation similar to T = 2??(I/MB), where I is the moment of inertia and MB represents the product of the magnetic moment and the applied magnetic field.

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to angular acceleration about an axis and depends on the mass distribution relative to that axis. For a rigid body whose dimensions are altered (for instance, by cutting it into halves), the moment of inertia changes in a way that can affect the time period of oscillation. In many oscillatory systems, I enters under the square root and plays a central role in setting the oscillation period.

Magnetic Dipole Moment

The magnetic dipole moment of a magnet quantifies the strength and orientation of its magnetism. In oscillatory problems involving magnets, the dipole moment determines how strong the coupling is between the magnet and an external magnetic field, thereby influencing the restoring torque. Scaling the magnet’s size—by cutting it into parts—affects the magnetic moment proportionally, which in turn impacts the period of oscillation.

Scaling Laws in Oscillatory Systems

Scaling laws describe how physical quantities such as period, moment of inertia, and magnetic moment change when the size or mass of a system is altered. In the context of a magnet oscillating under a magnetic torque, if the dimensions of the magnet are changed, both the moment of inertia and magnetic moment will change following specific geometric relationships. Understanding these proportionalities is crucial in predicting how modifications to the physical dimensions will affect the oscillation period.

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