Question

The period of oscillation of a magnet in a vibration magnetometer is 2 seconds. The period of oscillation of a magnet of moment of inertia four times that of the first magnet is :
A. $1 \mathrm{sec}$
B. $4 \sec$
C. $8 \mathrm{sec}$
D. $0.5 \mathrm{sec}$

Answer 1

Hint: Period is the amount of time a particle needs to complete an oscillation. The reciprocal of the frequency can be used to determine the oscillation's frequency. Here, in this problem we are going to use the expression of time period of a magnet in a vibration magnetometer.

Formula used:
The expression of time period of a magnet in a vibration magnetometer is,
$\mathrm{T}=2 \pi \sqrt{\dfrac{\mathrm{I}}{\mathrm{MB}}}$
Here, $I$ is the moment of inertia, $M$ is the magnetic moment and $B$ is the magnetic field.

Complete step by step solution:
A magnetic moment, also referred to as a magnetic dipole moment, is a measurement of an object's tendency to align with a magnetic field. A vector quantity is the magnetic moment. The magnetic moment vector frequently aligns with the magnetic field lines when the objects are positioned in that way.

A magnet's magnetic moment is directed from its southern to northern poles. The magnetic field that a magnet creates is inversely related to its magnetic moment.
$M^{\prime}=4 \mathrm{~m}$
$M \rightarrow$ magnetic moment of $1^{\text {st }}$ magnet
$M^{\prime} \rightarrow$ magnetic moment of now magnet

Now the expression of time period of a magnet in a vibration magnetometer is,
$\mathrm{T}=2 \pi \sqrt{\dfrac{\mathrm{I}}{\mathrm{MB}}}$
Using the above expression we can write,
$\mathrm{T} \propto \dfrac{1}{\sqrt{M}} \\ $
$\dfrac{T^{\prime}}{T}=\sqrt{\dfrac{M}{M^{\prime}}} \\
\Rightarrow \dfrac{T^{\prime}}{2}=\sqrt{\dfrac{M}{4 M}}$
$\Rightarrow \dfrac{T^{\prime}}{2}=\dfrac{1}{2} \\
\therefore T^{\prime}=1 \,\mathrm{sec}$

Therefore, the correct answer is option A.

Note: The process of any quantity or measure fluctuating repeatedly about its equilibrium value in time is known as oscillation. A periodic change in a substance's value between two values or around its central value is another way to define oscillation. The mechanical oscillations of an object are referred to as vibrations. But oscillations also happen in dynamic systems, or better said, in every branch of research. Even the heart's pounding causes oscillations. Oscillators, on the other hand, are things that move about an equilibrium point.