question_answer

A uniform cylinder of length L and mass M having cross-sectional area A is suspended, with its length vertical, form a fixed point by a mass less  spring, such that it is half submerged in a liquid of density \[\sigma \] at equilibrium position. When the cylinder is given a downward push and released, it stats oscillating vertically with a small amplitude. The time period T of the oscillations of the cylinder will be:                                                                  [JEE ONLINE 25-04-2013]

A)        Smaller than \[2\pi \] \[{{\left[ \frac{\operatorname{M}}{\operatorname{k}+\operatorname{A}\sigma g} \right]}^{1/2}}\]

B)  \[2\pi \sqrt{\frac{\operatorname{M}}{\operatorname{k}}}\]

C)  Larger than \[2\pi {{\left[ \frac{\operatorname{M}}{(\operatorname{k}+\operatorname{A}\sigma g)} \right]}^{1/2}}\]

D)       \[2\pi {{\left[ \frac{\operatorname{M}}{(\operatorname{k}+\operatorname{A}\sigma g)} \right]}^{1/2}}\]