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A horizontal cylinder is divided into two chambers by a thin smooth heat insulating piston of mass \( m \) and cross section area \( A \) initially. When the piston is in equilibrium, pressure of gas in each chamber is \( P_{0} \) and volumes of the chambers are \( V_{0} \) and \( 2 V_{0} \) as shown in the figure. The ideal gases in the two chambers are kept at constant temperatures \( T_{1} \) and \( T_{2} \). The piston is slightly displaced from equilibrium position and then released. Then the time period of small oscillations of the piston is
(a) \( 2 \pi \sqrt{\frac{m V_{0}}{3 P_{0} A^{2}}} \)
(b) \( 2 \pi \sqrt{\frac{2 m V_{0}}{3 P_{0} A^{2}}} \)
(c) \( 2 \pi \sqrt{\frac{3 m V_{0}}{2 P_{0} A^{2}}} \)
(d) \( 2 \pi \sqrt{\frac{m V_{0}}{2 P_{0} A^{2}}} \)
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