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A hemispherical concave mirror of (mass m and radius L) is rigidly attached at the lower end of a hinged uniform rod of (mass m and length L) as shown. The system is free to rotate in vertical plane. At time t=0 the system is released at small angular position $(4^{0})$ from vertical line. A point object (O) is placed vertically below the hinge at distance 2L. (Consider only paraxial rays).
Choose the correct option(s).

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The time at which separation between object and its image will be minimum first time is $\frac{π}{2} \sqrt{\frac{3 L}{2 g}}$

The separation between object and its image will be minimum two times in one oscillation

At $t = π \sqrt{\frac{3 L}{2 g}}$ , the separation between object and image is $\frac{4 π}{45} L (a p p r o x)$

At $t = \frac{3 π}{2} \sqrt{\frac{3 L}{2 g}},$ the coordinates of image are $(0, - 2 L)$

answer is A, B, C, D.

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Detailed Solution

Time period of oscillations

$T = 2 π \sqrt{\frac{I_{H}}{2 m g \times d}}, I_{H} = \frac{m L^{2}}{3} + \frac{8}{3} m L^{2} = 3 m L^{2} d = L T = 2 π \sqrt{\frac{3 L}{2 g}}$

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