Question

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 $($spring constant $k)$ such that it is half submerged in a liquid of density $\sigma$ at equilibrium position. The extension $x_o$ of the spring when it is in equilibrium is:

• A. $\frac{Mg}{k}(1+\frac{LA\sigma }{M})$
• B. $\frac{Mg}{k}(1-\frac{LA\sigma }{2M})$
• C. $\frac{Mg}{k}$
• D. $\frac{Mg}{k}(1-\frac{LA\sigma }{M})$

Answer 1

The correct option is B $\frac{Mg}{k}(1-\frac{LA\sigma }{2M})$

FBD of body in half submerged condition.


At equilibrium condition,
$k{x}_0+F_B=Mg$
$\Rightarrow k{x}_0+\frac{\sigma gV}{2}=Mg$
$\Rightarrow k{x}_0+\frac{\sigma g\times AL}{2}=Mg$
$\Rightarrow k{x}_o=(Mg-\frac{\sigma gAL}{2})$
$\Rightarrow x_o=\frac{Mg}{k}(1-\frac{LA\sigma }{2M})$

Tips: Draw the FBD of body and balance the forces in vertical direction.