Consider the following image from Wikipedia:

keeling boat with center of buoyancy, center of mass, and metacenter

The article states: "Whenever a floating body in a liquid is given a small angular displacement, it will start oscillating about some point, called the metacentre." I take this to mean that if I put a dot of paint at the metacenter of the boat, then as the boat oscillates, the dot of paint would be perfectly stationary (except that it would rotate in place).

In that case, as the boat oscillates, the center of mass moves left and right. This means that there must be net horizontal forces on the boat moving the center of mass left and right.

My understanding of the forces is:

  1. The gravitational force on the boat can be modeled as a single force acting downward at the center of mass.
  2. The buoyant force on the boat can be modeled as a single force acting upward at the center of buoyancy.
  3. There are no other forces considered in this model.

However, if the only forces on the boat are gravity, pointing down, and buoyancy, pointing up, there are no left or right forces. Nonetheless, the article implies that the center of mass moves left and right.

How is this possible?

  • $\begingroup$ You're examining a dynamic system with a static analysis. When rotating, water is moved and that creates (horizontal) forces. $\endgroup$ – BowlOfRed Feb 7 '19 at 17:05
  • $\begingroup$ @BowlOfRed Why do you think that? Do you have a source that calculates these forces, for example? Or one that explains physically why the metacenter should be stationary? $\endgroup$ – Mark Eichenlaub Feb 7 '19 at 17:38
  • $\begingroup$ No I have no other references. I just think that the static analysis is insufficient. It appears the mass of the ship is only part of the system. You should also consider the mass of the water that is displaced from one side to the other during the swing. $\endgroup$ – BowlOfRed Feb 7 '19 at 17:42
  • $\begingroup$ I came on the question when reading a source which explicitly said not to do that. ipho.org/problems-and-solutions/1995/… $\endgroup$ – Mark Eichenlaub Feb 7 '19 at 18:08

Sorry for my english. It's OK for equations but for text, it's difficult !

To define the metacentre, we limit ourselves to movements such that the volume of displaced water remains constant ("Isocarene" movement in French, I do not know how to translate it). In this type of displacement, Archimedes thrust always balances the weight. For a small inclination of the boat, if we limits to the first order in the inclination, the intersection of the vertical passing through the center of thrust is a fixed point of the boat: the metacentre.

When we establish the period of small oscillations of the boat, we make a number of approximations. We suppose first that we can apply the laws of static (Archimedes' thrust) to the moving ship. It is surely approached! But in this case, there is no horizontal force. The center of gravity remains on a vertical axis.

Movements are also considered such that the displaced volume remains constant. In this case, weight and Archimedes thrust compensate each other and there is no vertical force: the center of gravity remains fixed and the boat turns around the center of gravity. The metacentre oscillates around the center of gravity. Under these conditions, it is easy to establish the oscillation period.

The question arises as to whether one can have a fixed displaced volume and a fixed center of gravity. I think that strictly speaking it is not possible but at first order in the angle of rotation, the change in volume is zero and the hypothesis is consistent.

As the formula that gives the period of oscillation is called Bouguer's formula, I went to see Bouguer's book "Treaty of the ship, its construction and its movements" 1762 https://archive.org/details/bub_gb_lh1ZBtRvAb0C/page/n6 (The translation is from me and it's old French that I sometimes struggle to get in shape!)

Third section, Chapter I: From the point around which the vessel oscillates, which is called roll, and the part that gravity has in these oscillations. (p 369 ....)

"The problem is solved, it is no longer possible to doubt that it is around its center of gravity that the ship makes its oscillation." ... "It must be remarked that we neglect here the resistance which the water makes to the swaying of the ship; just as the resistance of the air to the movement of pendulums is usually neglected. This resistance is as null, compared to the other forces we consider, because no matter how large the oscillations of the ship, it has, because of the figure, that little water to move and that it does not shocked her with rather little speed. It is still assumed that the alternative inclinations are not large enough, so that the metacentre changes substantially in height relative to the center of gravity. "

In 1762, he problem is clearly stated!

Hope it can help !

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I agree with Vincent's answer, and am writing just to describe what I learned about this problem.

The statement on Wikipedia is incorrect (I've since deleted it); a ship does not oscillate about its metacenter. In the simple model of ship rolling, the only forces on the ship are the buoyant force and gravity, both of which are vertical. It is generally assumed that these forces are equal in magnitude, so that the center of mass of the ship doesn't move. The ship oscillates about its center of mass.

However, some sources erroneously state that the ship oscillates about its metacenter. I can speculate that this occurs for two reasons:

1 - Sources which describe the motion accurately sometimes state that the metacenter "is a fixed point" or "doesn't move", etc. What they mean is that as the ship rolls left or right, its metacenter is always at the same point on the ship. For example, if on a certain ship, the meta-center is in the center of the mast, 20 meters above the deck when the ship is rotated 1 degree, then when the ship is rotated two degrees, the metacenter is still in the center of the mast, 20 meters above the deck. However, its physical location in space has moved due to the ship's rotation.

2 - The buoyant and gravitational forces form a couple (a torque with no net force). The torque from a couple is independent of the origin chosen, and so the torque can be computed about the metacenter, even though the torque about the center of mass is desired.

3 - Few sources seem to explicitly state that the center of mass doesn't move, although this is implied in the analysis of those sources which analyze the motion correctly.

For example, a paper by Mungan and Emery derives the period of oscillation of a heeling ship using the moment of inertia about the center of mass, not the moment of inertia about the metacenter. I refer to that paper for clear definitions of the terms involved.

The problem from the International Physics Olympiad which I was reading when I encountered this question is also incorrect, as is the official solution to that problem.

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In this case there are no horizontal forces necessary to produce a torque, and flaudemus is correct that it is the torque that counts. You can create a torque with only uneven upward forces. The question seems to be why is rotation not about the center of mass. The reason is due to the buoyancy's requirement that the ratio of the object's volume above and below the water level remain constant assuming the object is not bouncing.

When the object is tilted, the center of mass has to rise in order to keep the volume of the object above water constant. You can pick an object like a cubic block an do a simple calculation to show this, but it is easy to visualize the case where you tilt the object enough to raise the center of mass to the level of the surface of the water.

When the object is released, the center of mass will fall causing the oscillations while the buoyancy force will maintain its height above water. The will be some variation in height due to the acceleration of the center of mass. So obviously the center of mass has to move to oscillate. If the object rotated about its center of mass, it would not oscillate; it would just spin.

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    $\begingroup$ Does the ship's center of mass move laterally during this oscillation? $\endgroup$ – BowlOfRed Feb 8 '19 at 23:29
  • $\begingroup$ Same question as BowlOfRed. I find it unclear what the motion of the ship is under your description. According to the passage I quoted from Wikipedia, the ship oscillates about the metacenter for small oscillations. In this case, the center of mass doesn't move up and down to first order; it only moves left and right to first order. $\endgroup$ – Mark Eichenlaub Feb 8 '19 at 23:37
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    $\begingroup$ It moves just like the center of mass of a pendulum rod. Up and down and left and right. For small oscillations up and down is relatively small, but just like a pendulum, the falling of the c-o-m when it is displaced causes the oscillation. The pivot point obviously has to be above the c-o-m. $\endgroup$ – Bill Watts Feb 9 '19 at 0:09
  • $\begingroup$ Then what horizontal forces push the center of mass left and right? $\endgroup$ – Mark Eichenlaub Feb 9 '19 at 0:31
  • $\begingroup$ There no horizontal forces. It is the torque that does it. If you twist one end of a rod, you apply a torque which moves the c-o-m without horizontal or vertical forces. The weight of the displaced c-o-m applies a torque as well as the uneven buoyancy force vertically pushing up. $\endgroup$ – Bill Watts Feb 9 '19 at 0:37

Your points 1.-3. are exactly correct. It is also important to recognize that these two forces are external to the floating body. As such they could in principle exert a net force on the floating body, which would change its linear momentum in time, and they could exert a torque (moment of force), which would change its angular momentum in time.

The second important ingredient is, that the floating body is rigid, i.e., it does not deform, an it has an inertial moment.

Concerning the net force, it will be zero in both cases, the left and the right figure. The buoyancy force acts vertically up, and the gravitational force acts vertically down. The floating body will sink into the water until both forces are of the same magnitude. Since they act in opposite directions, the net force is zero, and the linear momentum of the floating body is constant in time, i.e., it is zero.

What differs between the two figures is the magnitude of the torque. In the left case, the two external forces act along the same vertical axis, and the resulting torque is therefore zero. In case of the right figure, the buoyancy force, which acts vertically upwards from point B is not collinear with the gravitational force, which acts vertically downwards from point G. The result is a torque that acts as to restore the original equilibrium position of the body shown on the left.

In order to bring the floating body into the tilted position shown on the right, some external agent had to do work against the torque that gradually built up. This means that the floating body has acquired some potential energy as compared to the equilibrium situation on the left. The external agent can achieve the tilt by applying an external torque without applying a net linear force.

Once this external torque is removed, the system starts to oscillate. The remaining physics is that of a pendulum. Starting from the tilted position, the floating body will be set into rotating motion towards the equilibrium position. Doing this, potential energy is gradually converted into kinetic energy. When the body reaches the equilibrium position, the torque will be zero, but the body has all the potential energy converted into kinetic energy of rotation. It will therefore swing through the equilibrium position, again converting kinetic energy into potential energy, until the former is zero. For sufficiently small oscillation amplitudes, this oscillation will be harmonic.

In a realistic system, this motion will suffer from friction, which gradually converts kinetic energy into heat flowing off into the surrounding water. This situation corresponds to a damped harmonic oscillator.

This principle is responsible for the stability of ships against tilt. A ship will be the more stable, the lower point G is. This is why modern yachts have a lot of heavy mass in their keel. The restoring torque will also depend on the shape of the floating body, which will play a prominent role, if the tilt becomes larger and larger.

A more elaborate treatment can be found here.

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  • $\begingroup$ This does not answer the question. Specifically, why does the wikipedia article say that the metacenter is stationary? In that case, the center of mass moves. Why does the center of mass move left or right? What are the horizontal forces? $\endgroup$ – Mark Eichenlaub Feb 7 '19 at 17:35
  • $\begingroup$ If the metacenter is stationary, I think it is clear from my explanation why the center of mass moves left or right. So the only open question is: why is the metacenter stationary, right? $\endgroup$ – flaudemus Feb 7 '19 at 19:30
  • $\begingroup$ How should I know? I have no idea what an explanation of why the metacenter is stationary would look like. Here is what I want to know: if the center of mass moves left or right, what forces cause that left-right motion? Your answer does not identify any left-right forces. $\endgroup$ – Mark Eichenlaub Feb 7 '19 at 19:53
  • $\begingroup$ Based on your feedback, I edited my answer and spelled out the appearance of the left-right forces more explicitly. However, the description of the situation in terms of the torque would certainly be preferred in physics. $\endgroup$ – flaudemus Feb 7 '19 at 22:24
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    $\begingroup$ Your answer is incorrect. Vertical forces cannot be combined to create a net horizontal force. $\endgroup$ – Mark Eichenlaub Feb 7 '19 at 22:27

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