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Two small magnets have their masses and lengths in the ratio 1:2. The maximum torques experienced by them in a uniform magnetic field are the same. For small oscillations, the ratio of their time periods is
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The correct answer is:
$\frac{1}{2 \sqrt{2}}$
Let the mass of the first magnet $=m$
$\therefore$ Mass of second magnet $=2 \mathrm{~m}$
The time-period of oscillation of the magnet
$T=2 \pi \sqrt{\frac{1}{M B_H}}$
where, $I=$ Ml of magnet about the axis passing through its centre
$B_H=$ Earth's magnetic field
$M=$ Magnetic moment of magnet
$\therefore \frac{T_1}{T_2}=\sqrt{\frac{I_1}{M_1} \times \frac{M_2}{I_2}}$
Let length of the magnet $=L$
$\mathrm{Ml}$ of the magnets about the axis passing through one of the ends is $\frac{m L^2}{3}$
$\begin{aligned}
& \therefore I_1=m L^2 / 3 \text { and } I_2=\frac{(2 m)(2 L)^2}{3} \\
& \frac{T_1}{T_2}=\sqrt{\frac{m L^2 / 3}{M_1} \times \frac{M_2}{\frac{2 m \times(2 L)^2}{3}}}
\end{aligned}$
It is given that maximum torques experienced by the magnets are same
$\begin{aligned}
& \therefore & M \cdot B_H & =M_2 B_H \Rightarrow M_1=M_2\left[\because M_1=M_2 \text { (given) }\right] \\
\Rightarrow & & \frac{T_1}{T_2} & =\frac{1}{\sqrt{8}}=\frac{1}{2 \sqrt{2}}
\end{aligned}$
$\therefore$ Mass of second magnet $=2 \mathrm{~m}$
The time-period of oscillation of the magnet
$T=2 \pi \sqrt{\frac{1}{M B_H}}$
where, $I=$ Ml of magnet about the axis passing through its centre
$B_H=$ Earth's magnetic field
$M=$ Magnetic moment of magnet
$\therefore \frac{T_1}{T_2}=\sqrt{\frac{I_1}{M_1} \times \frac{M_2}{I_2}}$
Let length of the magnet $=L$
$\mathrm{Ml}$ of the magnets about the axis passing through one of the ends is $\frac{m L^2}{3}$
$\begin{aligned}
& \therefore I_1=m L^2 / 3 \text { and } I_2=\frac{(2 m)(2 L)^2}{3} \\
& \frac{T_1}{T_2}=\sqrt{\frac{m L^2 / 3}{M_1} \times \frac{M_2}{\frac{2 m \times(2 L)^2}{3}}}
\end{aligned}$
It is given that maximum torques experienced by the magnets are same
$\begin{aligned}
& \therefore & M \cdot B_H & =M_2 B_H \Rightarrow M_1=M_2\left[\because M_1=M_2 \text { (given) }\right] \\
\Rightarrow & & \frac{T_1}{T_2} & =\frac{1}{\sqrt{8}}=\frac{1}{2 \sqrt{2}}
\end{aligned}$
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