Figure shows a solid uniform cylinder of radius `R` and mass `M`, which is free to rotate about a fixed horizontal axis `O` and passes through centre of the cylinder. One end of an ideal spring of force constant `k` is fixed and the other end is higed to the cylinder at `A`. Distance `OA` is equal to `(R)/(2)`. An inextensible thread is wrapped round the cylinder and passes over a smooth, small pulley. A block of equal mass `M` and having cross sectional area `A` is suspended from free end of the thread. The block is partially immersed in a non-viscous liquid of density `rho`.
If in equilibrium, spring is horizontal and line `OA` is vertical, calculate frequency of small oscillations of the system.
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Answer

A, B, D

Explanation

In equilibrium,
`T + F = Mg` …(i)
When the block is further depressed by `x`, weight `Mg` remains unchanged, upthrust `F` increases by `rho A xg` and let `Delta T` be the increase in tension.

If `a` is the acceleration of block then,
`Delta T + rho A xg = Ma`...(ii)
Restoring torque on the cylinder,
`tau = [(kx)/(2) (R)/(2) - Delta T R] = [(kxR)/(4) - (Ma - rho Axg)R]`
`(1)/(2) MR^(2)alpha = [(kR^(2)theta)/(4) - (MR alpha - rho A g R theta)R]`
or `(3)/(2) MR^(2)alpha = [(kR^(2))/(4) + rho A g R^(2)]theta`
or `alpha = (-[k/4 +rho Ag])/((3)/(2)M)theta`
Here negative sign has been used for restoting nature of torque
`:. f = (1)/(2pi) sqrt|(alpha)/(theta)|`
`= (1)/(2pi) sqrt ((k + 4rho Ag)/(6M))` .