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.

Explanation

As the cylinder is half submerged in a liquid of density` rho` so upthrust on cylinder `=A(L//2)rhog` . If l is the extension of spring in equilibrium position, then
`kl=Mg-A(L//2)rhog` ..(i)
If y is the displacement of cylinder from equilibrium position downwards, then restoring force
`F=[k(l+y)-{Mg-A(L//2+y)rhog}]`
`-[Mg-(AL)/(2)rhog+ky-Mg+A((L)/(2)+y)rhog]`
`[`From (i) `]`
`=-[ky+Ayrhog)=-(k+Arhog)y` ...(ii)
From (ii), `Fpropy` and `-ve` sign show that the force F is directed towards the mean position, hence the cylinder if left free will execute SHM. Comparing
(ii) with the relatin, `F=-Ky,` we have
spring factor, `K=(k+Arhog)`
Inertia factor `=` mass of cylinder `=` M
Frequency of oscillation, `v=(1)/(2pi)sqrt((spri ng fact o r)/(i n e rtia fact o r))`
or `v=(1)/(2pi)sqrt((k+Arhog)/(M))`