Have you ever wondered why the
passenger basket on a hot air balloon is suspended underneath and not
simply strapped to the top of the balloon? After all, most hot air
balloons are used for sightseeing and a basket on the top would give
much better views.
Consider the balloon shown in the
left-hand diagram of below:
The balloon floats because the hot air
inside it is less dense than the cold air surrounding it, giving rise
to a buoyancy force acting upwards through B. When this force equals
the total weight of the balloon and basket, acting through the centre
of gravity G, the balloon will float at a constant altitude.
As the wind changes and the occupants
of the basket move around, the balloon will rock through a small
angle θ. Since the centre
of buoyancy is higher than the centre of gravity, any angular
displacement produces a turning moment which acts to restore the
balloon to an upright position. Such an arrangement is said to be in
stable equilibrium.
Now look at the bizarre case in the
right-hand diagram. The buoyancy force again equals the weight, but
here any angular displacement causes a turning moment which makes the
basket topple over. The reason for this is that the centre of
buoyancy B is below G. The situation is known as unstable
equilibrium.
Something very similar applies to
ships, but there are cases where stable equilibrium can be achieved
even where the centre of buoyancy is below the centre of gravity.
This occurs because the shape of the displaced water alters as the
ship rocks and so the centre of buoyancy moves sideways in the same
direction as the ship is leaning.
Therefore the line of action of the
buoyancy force also moves to the side of the ship which is further
down in the water, and the buoyancy force tries to lift the ship back
to the upright position. Whether or not the restoring moment is
enough to make the ship stable depends on the position of the point
where the line of action of the buoyancy force crosses the centreline
of the ship, known as the metacentre, M .
The distance between G and M is known
as the metacentric height.
If M is above G then the metacentric
height is positive and the ship is in stable equilibrium. If G is
above M then the metacentric height is negative and the ship is in
unstable equilibrium. This is the situation which led to the sinking
of King Henry VIII’s flagship, the Mary Rose, off Portsmouth.
This had sailed successfully for a
number of years and was just stable as it cast off on its fateful
last voyage, even though an unusually large shipment of weapons and
soldiers had raised the centre of gravity to danger level.
Finally, when the soldiers crowded up
onto deck for a last glimpse of land as the ship put out to sea, the
centre of gravity rose so high that the first big wave they
encountered away from the shelter of the harbour caused the ship to
topple completely over.
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