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In a Victorian society, an engineer has come up with a clever boat lift for his canal, using counterweights. In order to work correctly and most efficiently, the operators will need to know how much each boat or barge actually weighs.

How could he do this? Clearly the obvious answer is to put the boat into a tank and see how much the water level rises; but the point of this system is to weigh the vessel for lifting - so any solution that involves picking the ship up isn't practicable.

Plimsoll lines could work; but not all boats would have them (and you'd need to establish how much the boat actually weighs before you can accurately apply these anyway.

The best I've been able to come up with so far is a dry-dock - float the ship in, empty the water out completely and then fill it up again with a specific amount; then measuring how high the water is should give an answer. But it's not a very elegant (or fast) solution - can anyone do any better?

Edit to answer some of the comments: This is a 'dry' boat lift, it which the vessel sits on blocks (similar to those in a dry dock) and is lifted out of the water; hence the need to know the mass (rather than a 'wet' lift where the mass would always be the same).

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    $\begingroup$ If you're talking Victorians then you're missing their style of engineering. They didn't know how to calculate the maximum load something could take so they massively over engineered everything, and I do mean massively. The railway bridges and tunnels they built are still well within their tolerances. $\endgroup$
    – Separatrix
    Commented Dec 26, 2016 at 10:35
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    $\begingroup$ To expand on Separatrix's comment, Victorian engineers were fully aware of the need to make their works as future-proof as humanly possible: they overengineered their solutions even when they could calculate the required capacity. For example, when Joseph Bazalgette designed the sewerage system of London he made sure that it was expandable and that the main collector sewers would cope with a city of 4 million people, although in his time there were only 2 million. $\endgroup$
    – AlexP
    Commented Dec 26, 2016 at 12:50
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    $\begingroup$ As several of the answers say, for the purpose of a boat-lift in a canal, this is a non-question, you lift a tank of water + boat, generally balanced by a tank of water (ideally + boats, or you're wasting capacity) heading the other way. So the weight of the boat is compensated by the weight of water it displaces. $\endgroup$
    – user_1818839
    Commented Dec 26, 2016 at 17:27
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    $\begingroup$ It does not need to be a dry lift. See Panama Canal. $\endgroup$
    – sampathsris
    Commented Dec 27, 2016 at 9:41
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    $\begingroup$ Isn’t your lift also a scale? Attach your lift to the hull of the ship. Gradually add counterweight. For every ton of counterweight you add, the ship will rises a little out of the water. Eventually the entire ship is hovering out the water, and you add a little more counterweight to complete the hoist. The amount of counterweight when the ship is hovering (divided by whatever mechanical advantage you have) is the ship’s weight, not that anyone really cares, unless it’s used to assess the toll. $\endgroup$
    – Michael Zuschlag
    Commented Dec 28, 2016 at 15:06

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You describe a lift that uses counterweights, suggesting you want to weigh the ship to know how much counterweight to use. Depending on your lift design, that may not be necessary. Attach your lift to the hull of the ship. Gradually add more counterweight. For every ton of counterweight added, the ship will rise a little out of the water (equal to reducing its water displacement by one ton). The operators keep adding counterweight until the ship is out of the water, and thus have exactly the counterweight needed. They then can complete the lift (or lowering) by adding a little additional force (or weight adjustment) one way or the other.

To save time, the lift operators can use the methods suggested in AlexP’s answer to get a first estimate of how much counterweight to use, being careful to underestimate.

Your lift is also a scale. When the entire ship is hovering out the water, the amount of counterweight (divided by whatever mechanical advantage you have) is the ship’s weight. The operators don’t need to know that to figure counterweight –they already have that. However, it might be useful for other purposes, such as assessing the toll. The operators will also know from the engineer that a certain amount of counterweight (and thus weight of the ship) will break the lift, and will be careful not to put that much on.

(This is an elaboration of my comment, which the OP asked me to convert into an answer)

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The Victorian engineer--say for example Mr. Isambard Kingdom Brunel--would of course be interested in having a method which would give the exact weight of a ship; but in practice he would settle for a method which gave an approximate result, provided the result was not too wrong, because he would need to design his installation with a considerable factor of safety anyway. In his time they has some empirical formulas which gave an idea of the volume of the ship; similar formulas combined with the ship's draft could have been derived in order to estimate the displacement and by consequence the weight of the ship.

However, the entire question is based on questionable assumptions. Ships were not, and are not, designed and built so that they can be lifted out of the water. If one tries to lift a ship out of the water it will break unless a lot of care is taken with the supports. If you look at a ship in a dry dock you will notice the it needs to be supported over the entire length of the keel -- the keel is nowhere near the strength to resist being supported on a small number of points.

The Victorians did build a number of boat lifts, such as the lift on the Dorset and Somerset Canal or the Anderton Boat Lift. These work by lifting the ship while it is floating in a caisson; the entire question of precisely calculating the weight of the ship is circumvented by the expedient of adding or removing water until the up-going and down-going caissons are in equilibrium, that is, the water rises to the same level.

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    $\begingroup$ Further to the second paragraph, even in water it is necessary to take care when loading or unloading a modern cargo ship or tanker, to avoid breaking its back. $\endgroup$
    – David Aldridge
    Commented Dec 26, 2016 at 17:33
  • $\begingroup$ Excuse me, but aren't ships built on dry land rather than in the water? So obviously they knew how to support a ship. $\endgroup$
    – Mr Lister
    Commented Dec 27, 2016 at 14:07
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    $\begingroup$ @MrLister: They are built on carefully made and fitted supports and are launched at the earliest possible moment, with building being completed once they are afloat. When ships come into dry dock they again are placed on carefully fitted supports (and they are of course unloaded). You just cannot simply hook a ship to a crane and lift it because it will break. Ships are really fragile things. $\endgroup$
    – AlexP
    Commented Dec 27, 2016 at 14:59
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This may seem unintuitive, but if the ship is being lifted while it is floated in water, then its mass may not matter. This is because if the container holding the water is always of the same depth, then the water plus the ship floating in it will always weigh exactly the same.

You only need to know the weight if you are lifting the ship on its own, or in a container in which the water depth varies.

Similarly unintuitive -- the quantity of water "used" by a ship or ships in transiting a lock system is always exactly the same, regardless of the number/shape/size/weight of the ships.

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    $\begingroup$ this is a good point if the lift is a wet lift then the weight of the ship is irrelevant. the counterweight only needs to be calibrated to one weight. kinda like an anderson lift. upload.wikimedia.org/wikipedia/commons/8/85/… $\endgroup$
    – John
    Commented Dec 26, 2016 at 16:04
  • $\begingroup$ It's a very good point - but this would be a 'dry' lift where the boat is out of the water. $\endgroup$
    – Matt Bowyer
    Commented Dec 26, 2016 at 23:27
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    $\begingroup$ @MattBowyer, 'dry' ship lift would not have any advantage at all—when the lift is properly counter-balanced, and that is easy with the wet kind, the engine only has to overcome the friction, so the extra weight of water does not matter much. And it can't be built anyway—you'd need tailored support for each ship, because they are too fragile to just hang of two rope loops or something like that. $\endgroup$
    – Jan Hudec
    Commented Dec 28, 2016 at 12:27
  • $\begingroup$ Of course it can be built - I know this because several have. en.m.wikipedia.org/wiki/Canal_inclined_plane $\endgroup$
    – Matt Bowyer
    Commented Dec 29, 2016 at 21:28
  • $\begingroup$ @MattBowyer I think it's currently possible for a particular subset of cases, which are (A) small ships/boats (B) river traffic. The latter is a bit obvious I guess, because in general you wouldn't need to lift a ship at sea to a different level, but the hull form of a river boat is also very different in that it is largely flat bottomed. But given enough Victorian over-engineering I should think that it could be made to work for sea-going vessels if they were designed for that ability. More mechanically complex than locks, though. $\endgroup$
    – David Aldridge
    Commented Dec 30, 2016 at 9:25
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'Weight' = displacement

As Archimedes discovered (Eureka!), the weight of anything floating in water is equal to the volume of the water displaced by that floating thing.

By the Victorian era, ship hulls, especially ones made of metal, were drawn out in sketches before they were assembled in real life. From the sketches, accurate measurements could be made. A little bit of integration would determine the volume of water displaced by a vessel when the water line came up to point x on the hull, usually something painted onto the hull.

In the US Navy, we still use those lines to estimate the mass of fuel that we have loaded onto and off of the ship, to determine if we are leaking fuel.

enter image description here

The lines are at both the front and the back of the ship, and you can use the markings front and back, in case the ship is trimmed poorly, that is, it is lower in either the front or the back than the other, to determine the total 'displacement.' And if you look up wikipedia articles on ships, their size is listed in displacement rather than weight.

So the Victorians could and did make tables for their ship captains, to tell them what the total displacement of the ship was based on draft. This is very important since it could be used in reverse...namely how much mass can you take onto your ship (fuel and cargo) before your draft exceeds x. If he were going into shallow waters, a ship captain would not want to overload his ship.

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    $\begingroup$ +1 - I wanted to post the same thing, with very similar image. Ship builders already provide us with a good way to know. of course - can you trust the builder? $\endgroup$
    – Mołot
    Commented Dec 27, 2016 at 10:38
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    $\begingroup$ @Mołot You can when a big lawsuit is coming the builder's way after one of his oil tankers runs aground :) $\endgroup$
    – kingledion
    Commented Dec 27, 2016 at 15:40
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    $\begingroup$ For accuracy in inferring the weight from the load lines, you need to know the density of the water, of course, which is what the load lines on the right do: TF = Tropical fresh, F = Fresh, T = Tropical (salt), S = Salt, W = I can't remember. $\endgroup$
    – David Aldridge
    Commented Dec 28, 2016 at 17:07
  • $\begingroup$ @DavidAldridge W is winter. There is also a WNA for the North Atlantic, on some ships. The Navy doesn't use those marks, and for the ships I have been on at least, there was no distinction in the displacement tables between different water types and temperatures. I believe those are load limits for maximum loading, I don't think they are used in calculating actual displacement. $\endgroup$
    – kingledion
    Commented Dec 28, 2016 at 18:34
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    $\begingroup$ @kingledion Ah yes, winter. I guess that navies don't really have the same sort of range of displacements as cargo ships, either. $\endgroup$
    – David Aldridge
    Commented Dec 28, 2016 at 19:31
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Something similar to your counterweights solution has been done in real life here: https://en.wikipedia.org/wiki/Falkirk_Wheel

It's a rotating structure that holds two identical pools of water in each of its extremities. When a boat navigates into one of such pools, it'll dislocate an amount of water whose weigh is identical to its own, which makes the lift perfectly balanced without having to run any calculations or estimations on the ship's weight.

It can even have one boat coming up and one coming down at the same time.

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No need for a full dry dock,which is only really necessary if you're working on the hull of a boat. We already have a system that manipulates boats: canals.

If you have a canal structure such that the basic known depth allows a boat to float in, and then a pre defined volume of water is "flooded" into the canal lock, you can get total displacement without all the complications of a dry dock. A bonus to this system is that it can be integrated with the lift, so as soon as you calculate the counterweight you'd be able to engage the lift.

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    $\begingroup$ This won't work. Raising a ship in a lock changes only the depth of water under the keel, and the amount of water added depends only on the dimensions of the lock and the distance lifted. $\endgroup$
    – A. I. Breveleri
    Commented Dec 26, 2016 at 11:25
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    $\begingroup$ @A.I.Breveleri I'm not talking about RAISING the ship with the lock, I'm taking about WEIGHING the ship, allowing the calibration of the lift. The canal setup allows for floating the ship onto the supports needed to lift it, and the displacement gives approximate weight. $\endgroup$
    – Isaac Kotlicky
    Commented Dec 26, 2016 at 12:47
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    $\begingroup$ I thought you were talking about using a lock to determine the displacement of the ship. If you put supports under the ship and drain the water, you no longer have a lock, you have a dry dock. And when someone says "all the complications of a dry dock", that's what they mean: the supports. These supports need to be built and placed by someone who is expert in the particular ship's design, and thus could just tell you its displacement in the first place. $\endgroup$
    – A. I. Breveleri
    Commented Dec 26, 2016 at 20:10
  • $\begingroup$ Suppose your predefined volume is the volume that it would take to fill the lock without a ship by 1 foot, and you then float the ship in and add the water, the boat then rises by exactly 1 foot. What are you measuring to "get the total displacement"? $\endgroup$
    – Dave X
    Commented Dec 28, 2016 at 4:48
  • $\begingroup$ @DaveX you measure the difference of the displacements. In your example, there is no boat, as the boat displaced no water... let's say that the water line raised 3 feet when it should have raised 1 (a real example), then the boat is, effectively, as heavy as two feet of water inside the lock. Since the lock has predefined dimensions and water has a known density, the weight of the boat is Ldock * Wdock * 2 feet * density of water. $\endgroup$
    – Isaac Kotlicky
    Commented Dec 29, 2016 at 11:25
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Victorian engineers were capable of calculating the immersed volume of a hull. They did not do this very often because there was no need. For instance, Froude calculated how laboratory tests of ship hulls would scale up to real life.

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The obvious answer is to know the weight of each boat when built and know the weight of the cargo it holds, after all the cargo had to be loaded and balanced prior to sailing. I bet the captain of each ship had a pretty good idea of his ships displacement (or some other concept of weight that could be used to calculate the mass of the ship for lifting) that could simply be recorded in a ship log and given to the dockmaster. Since fine precision isn't necessary, a rough estimate would suffice.

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I don't know whether they'd be capable of such a calculation, but another possibility would be to apply a known force on the ship (mass) and watch how it's accelerating.

Place the ship somewhere such that a tower can be built (some distance away) in front of it (tower A) as well as another tower directly behind it (tower B).

Place a (heavy) mass at the top of tower A (mass A)

Place a (also heavy) mass at the bottom of tower B (mass B)

Use rope to tie the mass A via two rollers to the front of the waiting ship: one at the top of the tower, one at the bottom.

Do the same setup for tower B, but don't fix the rope to the back of the ship, just have someone there who can attach it to the ship on command.

Now let the mass A fall. This will accelerate the ship with a force $g * m_A - F_F$, over a time period $t_A$ until the mass reaches the ground.

ship accelerated due to mass A falling

Immediately when mass A hits the ground, give command to attach the other rope which is connected to mass B to the back of the (moving) ship. This will slowly decelerate the ship with a force of $m_B * g + F_F$ and lift mass B of the ground, until after $t_B$ the ship comes (briefly) to halt, before moving backwards.

ship halts due to mass B

Note the height $h_B$ that mass B was lifted maximally. Together with height $h_A$ which mass A was initially over the ground this can be used to calculate the ship's mass $m_S$. Start with the forces:

$m_S * a_A = F_A = g * m_A - F_F$

$m_S * a_B = F_B = g * m_B + F_F$

Now to keep things simple let's assume constant friction $F_F$ and thus constant acceleration in both cases (this will introduce a lot of error in the calculation, though).

Distance traveled under constant acceleration is $x = 1/2 * a * t^2$, thus

$a_A = 2 * h_A / t_A^2$

$a_B = 2 * h_B / t_B^2$

Calculate those two from the measurements.

Now back to the two equations regarding the forces. Solving one for $F_F$ one arrives at:

$F_F = g * m_A - m_S * a_A$

$m_S * a_B = g * m_B + F_F$

And thus:

$m_S * a_B = g * m_B + g * m_A - m_S * a_A$

And further:

$m_S = (m_B + m_A) * g / (a_B + a_A)$

As you might have guessed from the drawings, this is a quick and dirty sketch that I'm doing on my mobile phone. Thus everything above might be completely wrong.

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  • $\begingroup$ That is certainly an intriguing concept! Clearly hydrodynamics would have an effect; but it may well be that with similar boats that that makes little practical difference. $\endgroup$
    – Matt Bowyer
    Commented Dec 26, 2016 at 23:24
  • $\begingroup$ The hydrodynamics would definitely spoil this, I'm afraid. In addition to the hydrodynamic friction and wave losses, if you did this in a channel then you also have to deal with the flow of water from bow to stern, which causes the water level that the ship is in to lower, causing the ship to "squat". This means that channel dimensions and ship speed make an additional contribution to the calculation. $\endgroup$
    – David Aldridge
    Commented Dec 29, 2016 at 12:58
  • $\begingroup$ @DavidAldridge True, this won't do for an exact determination of the ship's mass. For that one would need to include these additional (and often non-constant, i.e. speed dependent) forces in the calculation. And additional measurements to determine unknown coefficients. But - and this is purely based on my experience as a hobby sailor, not on an uncertainty analysis - I think that my (relatively) simple method shown in my answer probably gets an answer for the ships mass within plus/minus 50kg of the real mass. $\endgroup$
    – Daniel Jour
    Commented Dec 29, 2016 at 13:15
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It is doubtful you will have a uniform weight on a particular boat that comes through your canal. Even the same ship on two trips will have different weights because of cargo, bilge, crew, and a lot of other factors. This leaves you with two options if you insist on the boat lift: Overengineering or pre-lift weighing.

Weighing

Other answers have gone into the specifics of how you'd obtain the weight of a vessel. The issue is that this takes time, and the longer it takes you to get a vessel through your canal the less money you are making, and the longer it takes to ship things. There's always the "just add weight to the counterweight bucket until it works" option, but that lengthens the time of the lift as well.

Overengineering

Overengineering is going to increase your construction costs, which is a big negative when you're trying to make money on your canal. It also would have bigger maintenance costs (as you have a bigger apparatus in general). You also have some major issues if you have a failure during a lift, which requires further overengineering of redundancies or accepting a certain level of risk.

Locks

The solution I would recommend to your canal owner is to just install a Lock in his canal. They have been used for millenia and the technology is not that complex (but the basic design can constantly be improved by technology. There's a reason we still build them). It circumvents your weight problem, because you just need to pump water in and let buoyancy take care of the rest. It also would be faster than a lift system requiring weighing beforehand (the drydock option requires more work than the lock in terms of pumping, and similar sized facilities), and is simpler and safer than the overengineered solution.

Even if you have an intricate pump system, and you suffer a failure of some kind, you can always operate the lock "manually" by opening the gates individually to flood and drain the lock. Or, if that's not an option, your fail-safe is to release the lower lock and then your boats are just "stuck" on the lower level canal until it can be repaired (which is far preferable to the "worst case" scenario of the lift, which is a boat shattered upon the ground).

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If you really MUST have a dry dock lifter, then I would recommend a solution that uses a combination of a test weight, and the amount by which the boat displacement changes.

Merely attached a fixed counterweight to the ship, observe the amount by which the boat rises in the water, and the weight of the ship can then be estimated based on the amount by which the water displacement of the boat was "negated"

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