A uniform cylinder of length (L) and mass (M) having cross sectional area (A) is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half - submerged in a liquid of density (rho) at equilibrium position. When the cylinder is given a small downward push and released it starts oscillating vertically with small amplitude. If the force constant of the spring is (k), the prequency of oscillation of the cylindcer is.

Options

A. `1/(2pi)sqrt((k)/(m))`
B. `(1)/(2pi)sqrt((karhog)/(m))`
C. `(1)/(2pi)sqrt((m+arhog)/(k))`
D. `(1)/(2pi)sqrt((k+arhog)/(m))`

Answer

D

Explanation

`kx + F_(B) = mg rArr kx = mg - rhoVg`
`kx = mg - (rhoAlg)/(2)`….(i)
Let cylinder be displaced through y then restoring force,
`f_("net") = -[k(x+y) + F_(B) - mg]`
`f_("net") = -[kx + ky + rhoA(l/2 + y) g - mg]`
`f_("net") = -[ky + rhoAgy] rArr ma = - [ky + rhoAgy]`
`rArr omega = sqrt((k + rhoAg)/(m)) rArr f = 1/(2pi)sqrt((k + rhoAg)/(m))`