A square metal pin with density $({}_{\text{p}\text{i}\text{n}})$ and length $L($into the paper$)$ is

floatIng on the surface of the water

Derive an expression for $b.$ You must take into account gravity, buoyancy and surface tension forces. For the surface tension force ignore the ends of the pin and assume that the water contact angle is such that it provides the maximum possible z$-$directed force.

Assuming ${}_{\text{p}\text{i}\text{n}}{}_{\text{w}\text{a}\text{t}\text{e}\text{r}}$ if the pin is pushed down such that $b$ is slightly larger than a will the pin stay on the surface of the water or sink$?$ Why or why not?

Detergent is gradually added to the water to lower its surface tension. Assuming ${}_{\text{p}\text{i}\text{n}}=9000k\frac{g}{m^3},{}_{\text{w}\text{a}\text{t}\text{e}\text{r}}=1000k\frac{g}{m^3}$ and $a=1mm$ calculate $$ right before the pin sinks$.$