$(1)$ An empty cylindrical bucket, $0\mathrm{.}3$ m in diameter and $0\mathrm{.}5$ m long whose wall thickness and weight can be considered as negligible is forced with its open end first into water until its lower edge is $4$ m below the surface. What force will be required to maintain position, assuming the trapped air to remain at constant temperature during the entire operation? Atmospheric pressure $\backslash (=1\mathrm{.}03\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$cm$\}^\{2\}\backslash ).$

$(20$ Marks$)$