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)[(K-Adg)/(M)]^(1//2)`
B. `(1)/(2pi)[(K+Adg)/(M)]^(1/2)`
C. `(1)/(2pi)[(K+Adg)/(M)]^(1//2)`
D. `(1)/(2pi)[(K-Adg)/(M)]^(1/2)`

Answer

C

Explanation

When the cylinder is given a downward push through a distance `y`, the restoring upward force in the spring `=-Ky.` Additional upward force due to buoyancy `=-Aydg.`
Net upward restoring force, `F=-(ky+Aydg)`
`=-(K+Adg)y` As, `F prop y` and it is directed towards equlibrium position, hence the cylinder will execute linear S.H.M.
Here, spring factor `=(KAdg)`
inertia factor`=` mass of cylinder `=M`
Frequency, `v=(1)/(2pi)sqrt ((spr i ng fact o r)/(i n ertia fact o r))`
`=(1)/(2pi)sqrt((K+Adg)/(M))`