A solid right circular cylinder of weight $10kg$ and cross sectional area $100c{m}^2$ is suspended by a spring, where $k=1kg-wt/cm$ and hangs partially submerged in water of density $1000kg/m^3$ as shown in the given figure. If it makes simple harmonic vertical oscillations with an angular frequency of $\sqrt{k}$, then the value of $k$ is___  (take $g=10m/s^2$ )

 

Let the cylinder be depressed a distance x meters into water from the position of rest. 

$K=1kg-wt/cm=100kg-wt/m$

Increase in tension of spring = $100x$ kg – wt
$(A=100c{m}^2={10}^{-2}{m}^2)$

Increase in buoyancy = $\frac{100}{{100}^2}x\times 1000kg-wt$

$=10xkg-wt$

Total unbalanced vertical force on the cylinder = $100x+10x=110xkg-wt$

Acceleration of cylinder = $\frac{110x\times 10}{10}m/s^2=110xm/s^2$

Angular frequency =$\sqrt{110} rad/s$