newtonian mechanics - Why do ships lean to the outside, but boats lean to the inside of a turn?

Small vessels generally lean into a turn, whereas big vessels lean out.

Why do ships lean to the outside, but boats lean to the inside of a turn?

For example, a boat leaning into a turn:

Image Source.

And a ship leaning out:

Image source

Answer

Your question became clear after you posted the images.

This corresponds to the difference in a turn as accomplished by a bike/motorcycle and a car/bus/truck. So let's study this case first.

During the turn within the accelerated reference frame there's an imaginary "centrifugal" force, which is directed outward the turn of course. OTOH the force applied by the ground has a component toward the turn (due to friction to prevent/reduce sliding). This creates a momentum that tends to lean the object outward the turn.

This is indeed what happens with most 4-wheel vehicles. They lean outward the turn, this transfers the pressure from the inner wheels to the outers, which causes appropriate change in the normal force applied to the wheels. This in turn creates a momentum which tends to lean the object toward the turn. By such the vehicle is leaned to some angle, after which equilibrium is achieved.

Now let's see what happens with 2-wheel vehicles (like bike). Since the normal force is applied in just 2 points, leaning outward the turn does not transfer the pressure, there's no change in the normal force, hence no momentum toward the turn is created. Moreover, leaning outward the turn displaces the mass from the 2-wheel axis, hence the gravitational force creates even more momentum outward the turn. The bike would just fall.

To accomplish the turn however, the biker leans the bike toward the turn deliberately. They do this by counter-steering, which involves microscopic inputs on the steering pushing the handlebars in the opposite direction to which they want to turn, thus leaning the bike into the turn. Displacing the mass causes gravitational force to create a momentum toward the turn. Which is in equilibrium with the momentum of the "centrifugal" force.

Now let's see what happens with vessels.

As with vehicles, the "centrifugal" force is applied outward the turn, the force applied by the water has a component toward the turn (due to the viscosity). Hence the "centrifugal" force's momentum is outward the turn. The difference is that there's a considerable part of the vessel under the water. Moreover, the center of mass is not required to sit above the water level. Another difference is that there are no discrete contact points with the water, instead water pressure is applied on all the underwater part of the vessel.

When the vessel leans (to either side) its configuration changes: its center of mass is displaced, its underwater part is changed, the volume and shape of the water "pushed out" changes as well. If the center of mass of the vessel + "pushed out" water raises - there's a momentum that tends to return the vessel back to its original state, hence it's stable.

Theoretically all the vessels are stable when at rest (otherwise they'd turn around). However during the motion some vessels are raised (like the small boat in your question) and become unstable. Such vessels definitely may not perform the turn unless deliberately leaned toward the turn. Simply because there's nothing to compensate for the "centrifugal" momentum. OTOH big vessels may remain stable even during the motion, with enough reserve to perform the turn as-is.

So, the factors to consider are:

• Vehicle configuration (sunk level, mass distribution, shape of the underwater part) during the motion.


• Required centripetal acceleration to perform the turn (velocity and radius).


• Exact forces imposed by the water (hydrodynamics).

Based on those one may see to which side the vessel leans during the turn.

There's however another interesting moment. If the vessel is unstable it should lean toward the turn. But how does this actually happen? Bicycle rider leans intentionally, otherwise he'd fall. He does it by counter-steering or displacing his own mass toward the turn, which is considerable WRT the mass of the bicycle.

But is this the case with the motor boats? I doubt if the mass of the rider is considerable WRT the mass of the boat. Plus, if this was the case, unskilled riders would turn around frequently, and I personally never saw this. There may be two explanations of this:

• Perhaps such boats are designed such that steering alone makes them lean toward the turn (due to a specific shape of the underwater tail, some hydrodynamical trick).


• During improper turn the vessel leans outward the turn, than it sinks a little, and in this new configuration there's an adequate momentum. So that the vessel doesn't turn around, it just passes the turn with a lower speed.