Hydrostatics: Buoyancy & Stability — Worked Examples

A homogeneous wooden cube has side length $0.500\ \text{m}$ and specific gravity $0.600$. Determine the submerged depth when it floats freely in water.

A $500\ \text{kg}$ concrete block has density $2400\ \text{kg/m}^3$. It is fully submerged in water and supported by a vertical cable from above. Determine cable tension.

A steel object has volume $0.0200\ \text{m}^3$ and density $7850\ \text{kg/m}^3$. Determine its true weight and the upward support force required to hold it fully submerged in water.

A rectangular pontoon is $8.00\ \text{m}$ long and $4.00\ \text{m}$ wide and weighs $400\ \text{kN}$. Determine its draft in seawater with specific weight $10.05\ \text{kN/m}^3$, assuming vertical sides over the immersed depth.

A rectangular pontoon is $10.0\ \text{m}$ long, $7.00\ \text{m}$ wide, and $2.50\ \text{m}$ deep. The pontoon weighs $700\ \text{kN}$ with $KG=1.25\ \text{m}$. A $300\ \text{kN}$ load has its center of gravity $3.00\ \text{m}$ above the deck. The pontoon floats in seawater with $\gamma=10.104\ \text{kN/m}^3$. Determine transverse $GM$.

The pontoon in the preceding example has displacement weight $1000\ \text{kN}$ and corrected $GM=1.070\ \text{m}$. A $50.0\ \text{kN}$ load is shifted $3.00\ \text{m}$ sideways. Estimate the equilibrium heel angle.

Ice has density $917\ \text{kg/m}^3$ and seawater has density $1025\ \text{kg/m}^3$. Determine the percentage of a freely floating iceberg below and above the water surface, neglecting entrained air and density variation within the ice.

A floating body displaces $100\ \text{m}^3$ of seawater of density $1025\ \text{kg/m}^3$. Its solid-loading metacentric height is $0.650\ \text{m}$. One partly filled freshwater tank has a rectangular free surface $4.00\ \text{m}$ long and $3.00\ \text{m}$ wide, with the $3.00\ \text{m}$ dimension transverse to the roll axis. Determine the free-surface correction and corrected $GM$.

Use the loaded pontoon from the metacentric-height example, initially $W=1000\ \text{kN}$ and $KG=2.525\ \text{m}$. Add $200\ \text{kN}$ of ballast with its center $0.300\ \text{m}$ above the keel. The rectangular waterplane remains $10.0\ \text{m}$ by $7.00\ \text{m}$ and seawater specific weight is $10.104\ \text{kN/m}^3$. Recompute draft and initial $GM$.

A floating body has corrected $GM=0.500\ \text{m}$ and mass radius of gyration $k=0.800\ \text{m}$ about its roll axis. Estimate its undamped small-roll period.

An empty rectangular tank has external dimensions $6.00\ \text{m}\times3.00\ \text{m}\times2.50\ \text{m}$ and is fully submerged below the groundwater table. The combined downward resistance from tank self-weight and justified permanent overburden is $600\ \text{kN}$. Using water specific weight $9.81\ \text{kN/m}^3$, determine buoyant uplift and the simple uplift safety ratio. Do not judge acceptability without a specified design criterion.