# How do you mathematically convert a ship's displacement of water to its weight on land? **URL:** \ **Category:** Factual Questions\ **Created:** [August 31, 2018, 5:23am UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556 "2018-08-31T05:23:34Z")\ **Posts on this page:** 11\ **Page:** 2
**Author:** ![Machine\_Elf](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@Machine\_Elf](https://boards.straightdope.com/u/Machine_Elf)\ **Post date:** [August 31, 2018, 2:56pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/21 "2018-08-31T14:56:44Z")
> [@SanDiegoTim](#): > > Does not the shape of the hull come into play? Would not two ships of the same actual weight displace a different volume of water if one was flat bottomed (like a barge) an the other a traditional “V” shaped hull? The pressure on the hull varies with depth below the surface of the water. If you have a barge with a broad, flat-bottomed hull, that broad, flat bottom won’t be sitting very deeply below the surface. Assuming vertical sides, the math is very easy: the buoyant force is the area of the flat bottom multiplied by the water pressure at that depth. As it happens, the water pressure at any depth is equal to the weight, per unit area, of a column of water of the same height. So, for example: Consider a barge, 50 x 100 feet. the flat bottom of the hull is one foot below the surface. How much does the boat weigh? 50X100X1 = 5000 cubic feet of water, times the density of water (62 pounds per cubic foot) = 310,000 pounds. The math gets more complicated for V-shaped hulls. Now, for each small bit of area on the surface of the hull, you have to consider: 1. the water pressure on that small bit of area (again, varies with depth) 2. the angle of that small bit of area with respect to horizontal With those two things figured, you can calculate the buoyant (upward) force on that small bit of hull area. It’s like thinking about each little tiny patch of that big, complicated hull as a separate, tiny flat-bottomed barge (albeit one that happens to be uncommonly short, narrow, and deep). Then you just add up the buoyant force for every small bit of area on the surface of the hull that’s under water. For shapes that can be conveniently described by a mathematical function, you can do these integrals by hand on paper. For arbitrarily complex hull shapes (e.g. real ships), computer software can execute the above procedure by brute force to give you the total buoyant force on the hull. The computer can also calculate the total weight of water that it would take to fill in the hole that would exist if the boat suddenly disappeared. And every time, the weight of that water will be equal to the buoyant force on the boat’s hull, regardless of shape. A V-shaped hull will penetrate more deeply into the water than a flat-bottomed barge, but it will also be narrower. The net result will be the same volume/weight of water displaced in either case. > [@mlees](#): > > > [@ZonexandScout](#): > > > > This is an effect that is quite noticeable for divers (SCUBA). As a diver descends, the air in the buoyancy compensator (or dry suit) is under greater pressure and decreases in volume. Since it displaces less water (which retains basically the same density at all depths), it becomes less buoyant and air has to be added to compensate for it. The opposite occurs when a diver goes toward the surface. Air has to be released periodically or the BC will expand dramatically and accelerate the movement upwards, which is usually not desirable. > > :eek: Do they have the help of gauges for that, or do they do it “by feel”? A full set of SCUBA equipment includes a depth gauge. During your ascent, you pay attention to that. If you’re ascending too rapidly (or not properly maintaining depth during a [decompression stop](https://en.wikipedia.org/wiki/Scuba_diving#Decompression)), you release air from your [buoyancy compensator.](https://en.wikipedia.org/wiki/Buoyancy_compensator_(diving)) The release valve is typically hand-operated; there’s also a fill valve that admits air from your breathing tank to increase buoyancy. (There’s also a tube to blow into it if your tank is empty and you’re trying to stay afloat on the surface.) During ascent you’re also supposed to NOT hold your breath. The air in your lungs wants to expand as you move to shallower depths, and if you don’t let it out, you can cause a [lung overpressure injury.](https://www.sportdiver.com/keywords/scuba-diving-certification/diving-doc-pulmonary-barotrauma) ---
**Author:** ![mlees](https://avatars.discourse-cdn.com/v4/letter/m/ba8739/32.png) [@mlees](https://boards.straightdope.com/u/mlees)\ **Post date:** [August 31, 2018, 3:49pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/22 "2018-08-31T15:49:40Z")
> [@Machine\_Elf](#): > > A full set of SCUBA equipment includes a depth gauge. During your ascent, you pay attention to that. If you’re ascending too rapidly (or not properly maintaining depth during a [decompression stop](https://en.wikipedia.org/wiki/Scuba_diving#Decompression)), you release air from your [buoyancy compensator.](https://en.wikipedia.org/wiki/Buoyancy_compensator_(diving)) The release valve is typically hand-operated; there’s also a fill valve that admits air from your breathing tank to increase buoyancy. (There’s also a tube to blow into it if your tank is empty and you’re trying to stay afloat on the surface.) > > During ascent you’re also supposed to NOT hold your breath. The air in your lungs wants to expand as you move to shallower depths, and if you don’t let it out, you can cause a [lung overpressure injury.](https://www.sportdiver.com/keywords/scuba-diving-certification/diving-doc-pulmonary-barotrauma) I knew about the depth gauge, but I was wondering if there was something that let you figure out how much air to pump in or out of your suit at any given depth. ---
**Author:** ![Just\_Asking\_Questions](https://avatars.discourse-cdn.com/v4/letter/j/ba9def/32.png) [@Just\_Asking\_Questions](https://boards.straightdope.com/u/Just_Asking_Questions)\ **Post date:** [August 31, 2018, 4:36pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/23 "2018-08-31T16:36:55Z")
> [@Sloe\_Moe](#): > > :smack::smack::smack: Eureka! 🆒 ---
**Author:** ![Machine\_Elf](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@Machine\_Elf](https://boards.straightdope.com/u/Machine_Elf)\ **Post date:** [August 31, 2018, 4:40pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/24 "2018-08-31T16:40:37Z")
> [@mlees](#): > > I knew about the depth gauge, but I was wondering if there was something that let you figure out how much air to pump in or out of your suit at any given depth. Not in my experience. The depth gauge was it, at least for ascents. If you couldn’t maintain your rate of ascent to your satisfaction, you just added/removed air until you were satisfied. When you’re swimming at a ~constant depth, there’s a range of buoyancy within which you can comfortably micromanage your depth using your fins, your hands, and your breathing (inhale = more buoyant, exhale = less buoyant). If your buoyancy is outside that range, then you end up working too hard to manage your depth, so you add or remove air from your BC as needed. ---
**Author:** ![Machine\_Elf](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@Machine\_Elf](https://boards.straightdope.com/u/Machine_Elf)\ **Post date:** [August 31, 2018, 4:44pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/25 "2018-08-31T16:44:34Z")
> [@Machine\_Elf](#): > > It’s like thinking about each little tiny patch of that big, complicated hull as a separate, tiny flat-bottomed barge (albeit one that happens to be uncommonly short, narrow, and deep). Then you just add up the buoyant force for every small bit of area on the surface of the hull that’s under water. I didn’t emphasize this section enough. It’s critically important: it’s the thing that tells you that for a given vessel weight, a V-hull will displace the same amount of water as a flat-bottomed hull. Afterall, if a V-hull (or any hull at all) can be thought of as a bazillion tiny flat-bottomed barges sailing very close together, it stands to reason that the relationship between total vessel weight and total volume of water displaced should be the same as it is for a flat-bottomed barge. ---
**Author:** ![Gray\_Ghost](https://avatars.discourse-cdn.com/v4/letter/g/c0e974/32.png) [@Gray\_Ghost](https://boards.straightdope.com/u/Gray_Ghost)\ **Post date:** [August 31, 2018, 5:33pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/26 "2018-08-31T17:33:16Z")
> [@robby](#): > > Good point. All of the previous discussion was, of course, in the absence of hydrodynamic forces… > > …Anyway, when a nuclear submarine is traveling submerged at high speed for an extended period of time (such as during a high speed transit across an ocean), it can be easy for this to all get out of whack, as the hydrodynamic forces become a much larger factor during the run. The personnel operating the sub can find that the sub is very heavy (or light) as it slows down, which requires quick corrective action. **robby** , if you can discuss this particular wrinkle, how much do changes in things that affect water density like salinity (like near the ice pack, or estuary/littoral operations), or temperature, affect hydrodynamic or hydrostatic concerns for buoyancy? Negligible, something that constantly has to be taken into account, or something in-between? Heck, do different currents as you ascend or descend in the water column materially affect buoyancy? Or are all of those factors accounted for in whatever the instruments tell you your position is, like the pressure gauge, depthfinder, or whatever sonar or other means you guys use to figure out where the ice roof is when operating in the Arctic? I knew that maintaining trim and attitude was really important for you guys, but I never understood that it could be that complicated until your posts. ---
**Author:** ![robby](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/robby/32/11048_2.png) [@robby](https://boards.straightdope.com/u/robby)\ **Post date:** [August 31, 2018, 6:19pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/27 "2018-08-31T18:19:43Z")
Wow, lots of posts since I was online this morning – I’m working my way down; sorry if a given question has already been addressed. > [@Machine\_Elf](#): > > Wouldn’t it be easy to know that this situation is developing? ISTM that the vessel’s net buoyancy could be estimated at any given speed by knowing the actual angles of the dive planes required to maintain a given depth, and comparing those actual angles to the angles that would be required if the vessel were in fact neutrally buoyant. A large discrepancy means a large excursion from neutral buoyancy; this knowledge would allow buoyancy correction to be applied before slowing to a speed at which the planes provide inadequate control authority. > > What am I missing? A couple of things. First of all, the effect of the control surfaces increases exponentially with speed. Second, the largest control surface on a submarine is the hull itself (referred to an an “up angle” or “down angle” of the ship), which is itself controlled by the diving planes (especially the stern planes). At high speed a small change in the angle of the ship (i.e. a fraction of a degree) can represent a tremendous amount of force which can mask a large deviation from neutral buoyancy. This is a much smaller effect at slow speeds (and disappears completely if the sub were to stop its forward motion). The greatly magnified effect of the control planes at high speeds is such a concern that limits are placed on the large control surfaces because it would be possible to put the sub into a dive that would exceed depth limits (possibly resulting in hull failure) before the dive could be recovered from. In some situations, one of the immediate actions for a stern plane failure in the down position is an emergency blow of the forward main ballast tanks. ---
**Author:** ![robby](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/robby/32/11048_2.png) [@robby](https://boards.straightdope.com/u/robby)\ **Post date:** [August 31, 2018, 6:44pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/28 "2018-08-31T18:44:08Z")
> [@mlees](#): > > I was reading a WW2 submariner’s autobiography, and he stated that as a submarine dove? dived? descended further into the ocean, the sub’s hull would be more and more compressed by the pressure of the water into a smaller volume, meaning that it now displaces less water, meaning… it’s buoyancy changes? Yes, as the saying goes in the submarine force, “The deeper you go, the heavier you get.” (I scuba dive, too, and the saying holds there as well.) More precisely, the effect is exactly as you state. The actual weight of the submarine is unchanged, but the buoyancy decreases as the submarine hull compresses, meaning that a neutrally buoyant sub becomes negatively buoyant as depth increases. > [@Xema](#): > > Definitely - there’s no way to make a perfectly rigid and incompressible submarine. > > It’s also true that as you descend, the density of the water increases - though not by much: Googling suggests a depth of 4km yields a density increase of about 1.8%. Which is ridiculously deep for non-research submarines. At typical military submarine depths, the density of water is essentially constant with respect to depth. The largest variable affecting water density is changes in salinity, with a smaller effect due to temperature. > [@mlees](#): > > …I also recall many of these books discussing how the crewman/officer responsible for keeping the sub’s trim may have to pump water from tank to tank as the center of gravity in the boat shifts (as crewman and cargo are shifted). > > It just occurred to me that if the trim officer had to compensate for the few hundred pound weights of two or three crewman moving aft, then the change in buoyancy due to depth may have to also addressed by this same man. Right, trim on submarines is traditionally controlled manually by the Chief of the Watch (COW) under direction of the Diving Officer of the Watch (DOOW) in the Control Room. (I believe the newest submarines may have completely or partially automated some of this.) Anyway, water is moved from variable ballast tank (VBT) to variable ballast tank, of which there are several. Water can also be pumped off the ship, or flooded in by use of valves. Anyway, on submarines that I served on, the movement of two or three crew members wasn’t a big deal, but the movement of a dozen or so crewmen was, as was movement of torpedoes (which weigh a couple of tons) or stores (e.g. food stores). Because the movement of a dozen or so crewmen caused a noticeable effect on the trim of the sub, a fun prank on a new DOOW or COW was to organize a so-called “trim party” with crew members walking back and forth from the torpedo room up forward to the engine room back aft. This could drive a new watchstander crazy. 😉 ---
**Author:** ![robby](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/robby/32/11048_2.png) [@robby](https://boards.straightdope.com/u/robby)\ **Post date:** [August 31, 2018, 6:46pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/29 "2018-08-31T18:46:26Z")
> [@ZonexandScout](#): > > This is an effect that is quite noticeable for divers (SCUBA). As a diver descends, the air in the buoyancy compensator (or dry suit) is under greater pressure and decreases in volume. Since it displaces less water (which retains basically the same density at all depths), it becomes less buoyant and air has to be added to compensate for it. The opposite occurs when a diver goes toward the surface. Air has to be released periodically or the BC will expand dramatically and accelerate the movement upwards, which is usually not desirable. > [@mlees](#): > > :eek: Do they have the help of gauges for that, or do they do it “by feel”? I’m a scuba diver, too. You do it by feel and experience. ETA: As **Machine Elf** states, you also have your depth gauge. Many people also have dive computers that tell them if they are ascending too fast. ---
**Author:** ![mlees](https://avatars.discourse-cdn.com/v4/letter/m/ba8739/32.png) [@mlees](https://boards.straightdope.com/u/mlees)\ **Post date:** [August 31, 2018, 7:00pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/30 "2018-08-31T19:00:22Z")
> [@robby](#): > > Because the movement of a dozen or so crewmen caused a noticeable effect on the trim of the sub, a fun prank on a new DOOW or COW was to organize a so-called “trim party” with crew members walking back and forth from the torpedo room up forward to the engine room back aft. This could drive a new watchstander crazy. 😉 And they say that “hazing” doesn’t happen anymore, in the modern Navy. Pull the other one… Anywho, thanks for the replies! ---
**Author:** ![robby](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/robby/32/11048_2.png) [@robby](https://boards.straightdope.com/u/robby)\ **Post date:** [August 31, 2018, 7:22pm UTC](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556/31 "2018-08-31T19:22:00Z")
> [@Gray\_Ghost](#): > > **robby** , if you can discuss this particular wrinkle, how much do changes in things that affect water density like salinity (like near the ice pack, or estuary/littoral operations), or temperature, affect hydrodynamic or hydrostatic concerns for buoyancy? Negligible, something that constantly has to be taken into account, or something in-between? Probably the best answer is “something in-between,” mainly because changes in salinity and temperature tend to happen relatively slowly. > [@Gray\_Ghost](#): > > Heck, do different currents as you ascend or descend in the water column materially affect buoyancy? Currents aren’t an issue for a submarine underway. One thing that _is_ an issue is the [sea state](https://en.wikipedia.org/wiki/Sea_state) of the surface of the ocean (i.e. how rough the water is on the surface). The rougher the seas, the more of a “suction” effect there is (due to the Bernoulli effect), which can sometimes cause an unwary submarine to “broach” (i.e. surface unintentionally) when going to periscope depth. This is embarrassing at best, and can be a safety hazard or operational hazard at worst. (Nuclear submarines are generally supposed to be submerged and undetectable when on station.) > [@Gray\_Ghost](#): > > …I knew that maintaining trim and attitude was really important for you guys, but I never understood that it could be that complicated until your posts. In all honesty, this stuff was refreshingly straightforward and simple when I learned it in the Navy’s Submarine School in Groton, Connecticut. Nuclear Power School, on the other hand, was far more difficult. I’d rather talk buoyancy any day instead of [nuclear reactivity](https://en.wikipedia.org/wiki/Nuclear_chain_reaction#Effective_neutron_multiplication_factor) and the [six-factor formula](https://en.wikipedia.org/wiki/Six_factor_formula). 😉 [Previous page](https://boards.straightdope.com/t/how-do-you-mathematically-convert-a-ships-displacement-of-water-to-its-weight-on-land/820556.md?page=1)