Flotation of Bodies & Conditions of Equilibrium

Flotation of Bodies:

It is a common observation that some bodies float in water while the others sink. For example- a log of wood and a ship made of steel float in water while an iron nail, a coin and a steel plate sink. Why does a particular body float or sink in a liquid can be explained as described below-

When a body is wholly or partly immersed in a liquid, it experiences a vertical upward thrust, T, equal to the weight of the liquid displaced. This upward thrust or the buoyant force, acting through the centre of buoyancy, opposes the weight of the body, ‘W’, acting downward through its centre of gravity. Thus, a body placed in a liquid is acted upon by two forces- (i) the upward thrust of the liquid (T) and (ii) the weight of the body (W) acting downward. Now, there are three possibilities-

(1) W > T, that is, the weight of the body is greater than the upthrust of the liquid. In this case, the body will experience a resultant force (W – T ) in the downward direction and will, therefore, sink.

(2) W = T, that is, the weight of the body is equal to the upthrust of the liquid. In this case, the resultant force (W – T) will be zero and the body will have no tendency to sink or rise. It will remain floating anywhere inside the liquid.

(3) W < T, that is, the weight of the body is less than the upthrust of the liquid. In this case, the body will experience a resultant force (W – T) in the upward direction and hence it will rise to the surface and float. The body will float with only a part is immersed in the liquid when the weight of the liquid displaced by its immersed portion is equal to its weight.

Conditions of Equilibrium of a Floating Body:

The following conditions are necessary for the equilibrium of a floating body.

(1) The weight of the body should be equal to the weight of the liquid displaced by it.

(2) The centre of gravity of the body and the centre of buoyancy of the displaced liquid should be in the same vertical line (law of flotation).

Though the above conditions are necessary for the equilibrium of a floating body but these are not sufficient for its stability. The stability of equilibrium is very essential for floating bodies like boat and ship.

Let us consider a boat as a floating body. When it is floating in an upright position, its centre of gravity (G) is above the centre of buoyancy (B₁). In this situation, the force due to gravity and the force due to upthrust is in the same vertical line (called the central line). Now when the boat tilts on one side due to some external disturbance, the shape of water displaced by it changes and so does the centre of buoyancy from position B₁ to new position B₂. At this stage, the force due to the weight of the boat and force due to upthrust of the liquid acting through (G) and B₂, respectively constitute a couple in the direction of the tilt. This tends to bring back the boat to its position of stable equilibrium, i.e. the untitled position. This will always happen when the metacentre (M) lies above the centre of gravity of boat or any other floating body. Thus for the stability of a floating body, the metacentre must lie above the centre of gravity of the body.

If the metacentre lies below the centre of gravity of the floating body, then the body is in a state of unstable equilibrium and consequently turns over.

Metacentre is a point at which the vertical line passing through the new position of centre of buoyancy intersects the line passing through the centre of gravity and the original position of the centre of buoyancy.