Two masses $m_{1} a n d m_{2}$ are connected by three springs of $k_{1}$ , $k_{2}$ and $k_{3}$ with the help of a right angle arm as shown below. The arm pivoted at point 'O' has a mass moment of inertia I. Which of the following expression represent the natural frequency of system?

$\sqrt{\frac{k_{1} a^{2} + k_{2} b^{2} + k_{3} c^{2}}{I + m_{2} a^{2} + m_{2} b^{2}}}$

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$\sqrt{\frac{k_{2} a^{2} + k_{1} c^{2} + k_{3} b^{2}}{I + m_{2} a^{2} + m_{1} c^{2}}} r a d / s$

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$\sqrt{\frac{k_{2} a^{2} + k_{1} c^{2} + k_{3} b^{2}}{I + m_{1} a^{2} + m_{2} b^{2}}}$

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$\sqrt{\frac{k_{1} a^{2} + k_{2} b^{2} + k_{3} c^{2}}{I + m_{2} a^{2} + m_{1} c^{2}}}$

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Solution

The correct option is B
$\sqrt{\frac{k_{2} a^{2} + k_{1} c^{2} + k_{3} b^{2}}{I + m_{2} a^{2} + m_{1} c^{2}}} r a d / s$

Total energy T.E of the system = kE + PE
$k E = \frac{1}{2} I θ^{2} + \frac{1}{2} m_{2} {(a θ)}^{2} + \frac{1}{2} m_{1} {(c θ)}^{2}$

$P E = \frac{1}{2} k_{2} {(a θ)}^{2} + \frac{1}{2} k_{3} {(b θ)}^{2} + \frac{1}{2} k_{1} {(c θ)}^{2}$

$Since in free vibration total energy of system remains consistent, i.e. \frac{d}{d t} (T . E) = 0$

$This gives, (I + m_{2} a^{2} + m_{1} a^{2}) θ + (k_{2} a^{2} + k_{1} c^{2} + k_{3} b^{2}) θ = 0$

$I_{e} ¨ θ + k_{e} θ = 0$

$ω_{u} \sqrt{\frac{k_{e}}{I_{e}}} = \sqrt{\frac{k_{2} a^{2} + k_{1} c^{2} + k_{3} b^{2}}{I + m_{2} a^{2} + m_{1} c^{2}}} r a d / s$

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