Floating Body Oscillation Point

When a body is floating in a liquid and is slightly tilted or displaced from its equilibrium position, it will oscillate. This oscillation happens around a specific point. To understand this, we need to consider the forces acting on the body: its weight acting through the center of gravity (G) and the buoyant force acting through the center of buoyancy (B).

When the body is tilted, the shape of the submerged volume changes. This causes the center of buoyancy (B) to shift to a new position (let's call it B'). The buoyant force still acts vertically upwards through this new center of buoyancy B'.

The metacenter (M) is a crucial point for determining the stability of a floating body. For small angles of tilt, the metacenter is defined as the point where the vertical line through the new center of buoyancy (B') intersects the original vertical line through the center of gravity (G) and the original center of buoyancy (B).

For a floating body to be in stable equilibrium, the metacenter (M) must be above the center of gravity (G). When the body is slightly displaced, the weight acting downwards through G and the buoyant force acting upwards through B' (whose line of action passes through M for small angles) create a restoring moment that tries to bring the body back to its original position. This restoring moment causes the body to oscillate.

The oscillation occurs around the point which is effectively the pivot for the restoring moment. For small angles of tilt, this pivot point is the metacenter.

Understanding Key Points

• Center of gravity (G): The point where the entire weight of the body is considered to act. It remains fixed relative to the body regardless of tilt.
• Center of buoyancy (B): The centroid of the submerged volume of the body. Its position changes when the body tilts because the shape of the submerged volume changes. The buoyant force acts upwards through this point.
• Metacenter (M): For small angles of tilt, it is the intersection point of the vertical line through the new center of buoyancy and the original vertical line. The stability of the floating body depends on the relative position of M and G.

Why Metacenter is the Oscillation Point

When the body is displaced, the restoring moment is created by the couple formed by the weight at G and the buoyant force at B' (acting vertically upwards through M). This moment tends to rotate the body back towards equilibrium. The point around which this rotation or oscillation effectively happens is the metacenter, especially for small displacements.

Consider the distance GM, known as the metacentric height. If M is above G (GM is positive), the body is stable and will oscillate. If M is below G (GM is negative), the body is unstable and will capsize. If M coincides with G (GM is zero), the body is in neutral equilibrium.

Therefore, a slightly displaced floating body oscillates about its metacenter.

Eliminating Other Options

• Center of buoyancy: This point shifts when the body tilts, so the oscillation is not about a fixed center of buoyancy.
• Center of pressure: This term is typically used when dealing with pressure distribution on submerged surfaces or in aerodynamics (on airfoils). It is not the point of oscillation for a floating body due to buoyancy effects.
• Center of gravity: While the weight acts through the center of gravity, the oscillation is not centered at G itself. The restoring moment is a couple involving G and B', pivoting effectively around M.

Thus, the correct point of oscillation for a slightly displaced floating body is the metacenter.