Mass on a Spring semi-submerged in a liquid

[MatinSAR]

Homework Statement

According to the figure, the metal cylinder with mass M and volume V is connected to a fixed point from above by a spring with spring constant k and it floats inside a liquid of density p so that half of its height is inside the liquid. What weight should we put on it so that two-thirds of the height is inside the liquid?

Relevant Equations

Fluid mechanics.

Hi , Can someone tell me wether my answer is corrrect or false ?

 

[erobz]

I'm getting something different from you. The difference is I define $d'$ in terms of $d$ and $h$ differently. Actually, they work out equivalently. If you just eliminate $Mg$ via your first equation, and put $L$ in terms of $d$ and $h$ , I think we will get the same result.

My end result is purely a function of $k,h,\rho,g,V$