$1.$ An $100$ mm diameter composite solid cylinder consists of an $100$ mm diameter, $40$ mm thick metallic plate having specific gravity $5\mathrm{.}0,$attached at the lower end of an $100$ mm diameter wooden cylinder of specific gravity $0\mathrm{.}75.$ Find the limits of the length of the wooden portion so that the composite cylinder can float in stable equilibrium in water $($specific gravity $1\mathrm{.}0)$ with its axis vertical.