Question

A magnetic needle of magnetic moment and moment of inertia is performing simple harmonic oscillations in a magnetic field of 0.01 T. Time taken for 10 complete oscillations is:
A. 8.76 s
B. 6.65 s
C. 8.89 s
D. 6.98 s

Answer 1

By using the equation for time period, we can solve this question.
$T=2\pi \sqrt{\dfrac{I}{MB}}$
Where $I$ is the moment of inertia, $M$ is the magnetic moment and $B$ is the magnetic field.

Complete answer:
In the question, it is given that
Magnetic moment is
$M=6.7\times {{10}^{-2}}$ $A{{m}^{2}}$
Moment of inertia is
$I=7.5\times {{10}^{-6}}kg{{m}^{2}}$
Magnetic field is
$B=0.01T$
We know that time period of oscillation is given by the formula
$T=2\pi \sqrt{\dfrac{I}{MB}}$
Applying the values given in the question to the equation gives,
$T=2\pi \times \sqrt{\dfrac{7.5\times {{10}^{-6}}}{6.7\times {{10}^{-2}}\times 0.01}}$
Performing the operations gives,
$\begin{aligned} & \\\ & T=2\pi \times 0.1058 \\\ & T=0.66477 \\\ \end{aligned}$
Therefore we will get like,
$T=0.6647s$
We know Time period is known as the time required to complete one oscillation.
Therefore, to complete ten oscillations
$\begin{aligned} & t=10\times T \\\ & t=10\times 0.66477 \\\ & t=6.6477 \\\ & \\\ \end{aligned}$
Therefore
$t\approx 6.65s$

So the correct answer is option B.

Additional Information:
Magnetic moment is the product of the pole distances with the strength of the magnetic pole. It is a vector quantity. Moment of inertia is the quantity which shows the reluctance of an object to acquire angular acceleration. In simple words it is the ratio of angular momentum of a body to the angular velocity around the principal axis of the body. Magnetic field is the tendency of attraction or influence on an electronic charge.

Note:
There is a chance of getting confused with the two options B and D as they are almost similar values.so be careful in using the decimal digits during operations like multiplication, division etc. Here the answer for this question is $t\approx 6.65s$.