Oscillations

A tank contains two immiscible liquids of densities 6 and 2 . The higher density liquid is filled up to a height L/2 from the bottom. A thin rod of density and length L is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is 2 n Lg, where g is the acceleration due to the gravity. The value of n is: [figure]

As shown in the figures, a uniform rod OO' of length l is hinged at the point O and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end (O') of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is f1. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is f2. Ignoring gravity and assuming motion only in the plane of the diagram, the value of f1f2 is: [figure]

1. 2
2. 2
3. 52
4. 25