Question

An aluminium wire is $m_{cork.}.$ The densities ${\rho }_{corck},{\rho }_{al}$ and ${\rho }_w$ of cork, aluminium, and water are $0.5\times {10}^{3}kg/m^3,2.7\times {10}^{3}kg/m^3,$ and $1\times {10}^{8}kg/m^3$ respectively.Determine the minimum mass $m_{al}$ of the wire that should be wound on the cork so that the cork with wire is completely submerge in water.

Answer 1

The condition of complete submergence of the body is
$M\ge {\rho }_{w}V$, where $M$ is the mass of the body, and $V$ is its total volume. In this case under consideration, we have
$M=m_{cork}+m_{al}$, $V=\frac{m_{cork}}{{\rho }_{cork}}+\frac{m_{al}}{{\rho }_{al}}$.
Hence it follows that the minimum mass of the wire is
$m_{al}=\frac{{\rho }_{al({\rho }_w-{\rho }_{cork})}}{({\rho }_{al}-{\rho }_w){\rho }_{cork}}{m}_{cork}\simeq 1.6m_{cork}$.