Buoyancy explains why ships, pontoons, and offshore platforms can float while carrying large loads. A floating object is supported by an upward force equal to the weight of the fluid it displaces. Engineers must also check stability, because floating is not enough if the vessel tips too easily.
The key idea is to compare where weight acts, where buoyancy acts, and how those points move when the hull tilts.
When a ship heels to one side, the underwater shape changes, so the center of buoyancy shifts sideways. The upward buoyant force through this shifted point and the downward weight through the center of gravity create a turning effect called the righting moment. The metacenter is a geometric point used to judge whether small tilts produce a restoring torque or an overturning torque.
A stable floating body has positive metacentric height, meaning the metacenter is above the center of gravity.
Key Facts
- Buoyant force equals the weight of displaced fluid: F_b = rho_fluid g V_displaced.
- For floating equilibrium, buoyant force equals weight: F_b = W = mg.
- Average density condition for floating: rho_object,avg < rho_fluid.
- The center of buoyancy B is the centroid of the displaced fluid volume.
- Metacentric height is GM = KM - KG, where G is center of gravity and M is metacenter.
- For small heel angles, righting moment is approximately tau = W GM sin(theta).
Vocabulary
- Buoyant force
- The upward force exerted by a fluid on an object equal to the weight of the fluid displaced by the object.
- Center of buoyancy
- The point through which the buoyant force acts, located at the centroid of the displaced fluid volume.
- Center of gravity
- The point through which the total weight of an object acts.
- Metacenter
- The point where the vertical line through the shifted center of buoyancy intersects the original centerline for a small tilt.
- Righting moment
- The restoring torque that tends to rotate a tilted floating body back toward upright equilibrium.
Common Mistakes to Avoid
- Confusing center of buoyancy with center of gravity is wrong because buoyancy depends on displaced water shape while gravity depends on mass distribution.
- Assuming a floating object is always stable is wrong because an object can float and still capsize if its righting moment is negative or too small.
- Using total object volume instead of displaced volume in F_b = rho g V is wrong for floating objects because only the submerged volume displaces fluid.
- Thinking a lower center of gravity always guarantees stability is incomplete because hull shape and the location of the metacenter also control the metacentric height.
Practice Questions
- 1 A rectangular pontoon displaces 12.0 m^3 of freshwater with density 1000 kg/m^3. What is the buoyant force on it? Use g = 9.8 m/s^2.
- 2 A small boat has weight W = 18,000 N and metacentric height GM = 0.80 m. If it heels by 10 degrees, estimate the righting moment using tau = W GM sin(theta).
- 3 A ship is loaded with heavy cargo high above the deck, raising its center of gravity. Explain how this changes GM and why it can make the ship less stable even if it still floats.