On the ghats of Varanasi, a boatman takes a family out on the Ganga at dawn. As the boat leaves the bank, two of them stand up to take photographs. The boat rocks, and the boatman at once asks everyone to sit down. He is not being fussy.
Two forces, two points
Two forces act on a floating boat. Its weight \(W\) acts down through its centre of gravity \(G\). The buoyant force acts up through the centre of buoyancy \(B\), the centre of gravity of the water the hull displaces. For a boat floating upright, both points lie on the boat's centre line, and the two forces act along one vertical line. For most boats \(B\) lies below \(G\), because only the lower part of the hull is under water.
Now tilt the boat to one side. The low side goes deeper and pushes more water aside, and the high side comes up and pushes less. The displaced water now has more of its volume on the low side. So its centre, \(B\), moves towards the low side. \(G\) does not move with respect to the boat, as long as nothing inside it moves.
A boat floating upright is tilted so that its right side goes down. What happens to its centre of buoyancy \(B\)?
The metacentre
Once the boat tilts, the weight and the buoyant force are still equal, because the boat still floats. But they no longer act along one line. Two equal and opposite forces that are not in line form a couple, which turns a body without moving it along. Whether this couple turns the boat back or further over depends on where the two lines lie.
Draw the vertical line up through the new \(B\). It meets the centre line of the boat at a point \(M\), called the metacentre. If \(M\) lies above \(G\), the buoyant force acts on the low side of the weight's line. The couple then turns the boat back upright. If \(M\) lies below \(G\), the buoyant force acts on the high side, and the couple turns the boat further over.
A floating body is in stable equilibrium when its metacentre lies above its centre of gravity. It is in unstable equilibrium when the metacentre lies below it.
Start with the cargo low in the wide hull, in the sim above. \(G\) is \(0.56\,\text{m}\) above the keel, and \(M\) about \(1.64\,\text{m}\). At a tilt of \(10°\) the two lines of action are \(0.19\,\text{m}\) apart, with the buoyant force on the low side. This horizontal distance is called \(GZ\). Now put the cargo on the deck. \(G\) rises to \(1.95\,\text{m}\), above \(M\). At \(10°\) the lines are \(0.05\,\text{m}\) apart the wrong way, and the couple turns the barge over.
The distance \(GM\), from \(G\) up to \(M\), is the metacentric height. For small tilts, \(M\) stays nearly fixed, and the arm of the couple is \(GZ = GM\sin\theta\). A large \(GM\) means a strong push back upright. In the sim, the wide hull with the cargo low has \(GM = 1.08\,\text{m}\), and \(1.08 \times \sin 10° = 0.19\,\text{m}\).
Wide and flat, or narrow and deep
A wide hull swings a lot of water from the high side to the low side when it tilts. So \(B\) moves far, and \(M\) lies high. A narrow hull moves little water across, and its \(M\) lies low. The narrow hull in the third chip has the same weight and \(G\) at \(0.60\,\text{m}\), but its \(M\) is only \(1.03\,\text{m}\) above the keel. At \(10°\) its arm is \(0.08\,\text{m}\), less than half the wide hull's \(0.19\,\text{m}\).
A boat of mass \(2000\,\text{kg}\) has a metacentric height of \(0.5\,\text{m}\). A gust tilts it by \(10°\). Take \(g = 10\,\text{m/s}^2\) and \(\sin 10° = 0.174\). Find the couple that turns it back upright.
The weight is \(W = 2000 \times 10 = 20\,000\,\text{N}\), and \(GM = 0.5\,\text{m}\).
\(\tau = W \cdot GM \sin\theta\).
\(\tau = 20\,000 \times 0.5 \times 0.174 = 1740\,\text{N·m}\).
A couple of about \(1.74\,\text{kN·m}\) turns the boat back. With the load raised until \(GM\) is \(0.1\,\text{m}\), it would be five times smaller.
Three barges of the same weight float in still water. Heights above the keel are given for \(G\) and \(M\). Rank them by how strongly they turn back upright after a small tilt, strongest first.
A fishing boat of mass \(1500\,\text{kg}\) has a metacentric height of \(0.4\,\text{m}\). Take \(g = 10\,\text{m/s}^2\) and \(\sin 6° = 0.105\). Find the righting couple when it is tilted by \(6°\), in N·m.
Why is a boat with its cargo on the floor more stable than the same boat with the same cargo on the deck?
Under the water
A submarine under water is surrounded on every side. When it rolls, the displaced water keeps the same shape, the whole hull. So \(B\) stays at the centre of the hull's volume and does not move to the low side. There is no metacentre above \(B\) to help. The condition becomes simpler and stricter: a submerged body is stable only when \(B\) lies above \(G\).
In Fig 5.1, first put \(G\) \(0.3\,\text{m}\) below \(B\). A roll of \(30°\) then puts the two forces out of line, and they turn the hull back. The rolling dies away. With \(G\) \(0.3\,\text{m}\) above \(B\), even a roll of \(5°\) gives a couple that turns the hull further. It rolls right over and settles upside down, with \(G\) below \(B\). So a submarine carries its batteries and heavy machinery low, and a ship carries ballast at the bottom of its hull.
A floating ship can be stable with its centre of gravity G above its centre of buoyancy B. A submarine under water cannot. Explain the difference.
When a floating ship tilts, the low side displaces more water, so B moves to the low side. If the metacentre M stays above G, the buoyant force acts on the low side of the weight and the couple rights the ship, even with G above B. A submerged submarine displaces the same shape of water at any tilt, so B does not move. The only way the couple can turn it back is for B to be above G.
On 10 August 1628 the new Swedish warship Vasa set out on her first voyage from Stockholm. She carried two decks of heavy bronze cannon high in the hull, and too little ballast low down. Her centre of gravity was too high. Barely 1300 metres out, a gust of wind heeled her over. Water poured in through the open gun ports, and she sank in minutes.
Back on the Ganga, the boatman knows this physics. When passengers stand, the centre of gravity of the loaded boat rises towards \(M\). The metacentric height shrinks, and so does the couple that answers a tilt. A lurch that a seated party would ride out can then turn the boat over. Sitting low, with the load spread across the floor, keeps \(G\) low and the boat stable.
Cargo ships at a port are loaded to a plan. Heavy containers go low in the hold, and light ones high on deck. When a ship sails with little cargo, it pumps sea water into ballast tanks at the bottom of the hull. The water lowers \(G\), and a submarine uses the same tanks to sink and to rise.
- Treating the centre of buoyancy as a fixed point of the boat.\(B\) is the centre of the displaced water. It moves to the low side when a floating body tilts.
- Requiring \(B\) above \(G\) for every floating body.A floating body needs \(M\) above \(G\). Only a fully submerged body needs \(B\) above \(G\).
- Thinking that moving cargo changes the metacentre.\(M\) depends on the hull and the displaced water. Moving cargo moves \(G\).