At a certain place a magnet makes 30 oscillations per...

At a certain place a magnet makes 30 oscillations per minute. At another place where the magnetic field is double, its time period will be
(a) $\sqrt{2} \mathrm{~s}$
(b) $2 \mathrm{~s}$
(c) 45
(d) $\frac{1}{2} s$

Key Concepts

Frequency and Time Period Relationship

The frequency of an oscillatory system is defined as the number of oscillations per unit time and is the inverse of the time period. This fundamental relationship allows one to determine the characteristics of the motion; knowing either the frequency or the time period provides essential insight into the behavior of the oscillating system.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes a system in which the restoring force is directly proportional to the negative of the displacement from an equilibrium position. It is a fundamental concept in oscillatory systems and provides the framework for understanding periodic motions found in various physical scenarios, including the oscillations of a magnetic dipole in a magnetic field.

Magnetic Dipole Oscillations

Magnetic dipole oscillations occur when a magnet with a magnetic moment is subject to a torque in an external magnetic field, causing it to oscillate about an equilibrium position. The dynamics of this system can be analyzed using the principles of SHM, where key parameters such as the moment of inertia and the magnetic field strength determine the oscillation characteristics.

Dependence on Magnetic Field Strength

For a magnetic dipole undergoing small oscillations in a uniform magnetic field, the time period is inversely proportional to the square root of the magnetic field strength. This relationship implies that increasing the magnetic field results in a shorter oscillation period, as the stronger field increases the restoring torque acting on the dipole.

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