A thin magnetic needle oscillates in a horizontal plane with...

A thin magnetic needle oscillates in a horizontal plane with a period $\mathrm{T}$. It is broken into n equal parts. The time period of each part will be
(a) $\mathrm{T}$
(b) $\mathrm{n}^{2} \mathrm{~T}$
(c) $(\mathrm{T} / \mathrm{n})$
(d) $\left(\mathrm{T} / \mathrm{n}^{2}\right)$

Key Concepts

Magnetic Dipole in a Uniform Magnetic Field

A magnetic needle behaves like a magnetic dipole when placed in a uniform magnetic field. The magnetic moment interacts with the external field, producing a torque that tends to align the needle along the field direction. For small angular displacements, this restoring torque is proportional to the angular deviation, leading to oscillatory motion analogous to that of a simple harmonic oscillator.

Moment of Inertia Scaling

The moment of inertia, which depends on both the mass and the square of the distance from the axis of rotation, is crucial in determining the oscillation period of a body. When the length of the object is scaled by a factor and its mass is distributed accordingly, the moment of inertia changes by a power of that scaling factor. This concept is used to determine how the period of oscillation changes when the original needle is divided into equal parts.

Simple Harmonic Motion of a Physical Pendulum

A physical pendulum executing small oscillations exhibits simple harmonic motion, where the period depends on the square root of the ratio between the moment of inertia and the effective restoring torque. When both the moment of inertia and the magnetic torque (acting as the restoring force) change due to modifications in the body's dimensions and mass distribution, the period of oscillation adjusts accordingly.

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