Buoyancy: Archimedes' Wooden Spheres

[Kilt]

A number of questions have come up on canoe forums relating to the “buoyancy” of different woods out of which paddles can be made. The general issue has been to make the most “buoyant” paddle.

Generally, here’s the goal. In certain canoe strokes called in-water slice returns, you don’t lift the paddle out of the water after it is behind you at the conclusion of the stroke. Instead, you rotate the paddle 90 degrees while it is still immersed so the edge faces forward like a knife. You then slice the paddle forward through the water until it approaches the beginning position for the next stroke. To make the next stroke, you must expend energy to lift the paddle out of the water. The goal is to have the “buoyancy” of the paddle do most of that lifting work – so that the paddle “pops” up out of the water “on its own”. This can be an important energy saving technique for canoeists who take 70 strokes per minute for 10 hour marathon races. Or so they believe.

Some who have read some physics formulas say that heavier paddles actually are more “buoyant” because they displace more water and hence are opposed by a larger “buoyant force”. This seems intuitively wrong to many.

Others say “buoyancy” can be increased by adding volume to the paddle – e.g., making the blade thicker. Even if this is true, which many intuitively doubt, this increased blade thickness would have other (non-“buoyancy”) handling characteristics that are undesirable.

I think I have concocted a series of simple experiments with wooden spheres that could clear up this confusion. But not all of the answers are intuitively clear, and I do not have the expertise to actually do the experiments or manipulate the relevant equations. So … I am soliciting qualitatively and, preferably, quantitatively definitive answers.

Assume I have three perfect wooden spheres made of wood:

L1 – This is sphere made of Light (i.e., less dense) wood, 1 unit in volume.

L2 – This is a sphere made of the same Light wood, but 2 units in volume.

H1 – This is a sphere made of Heavier (more dense) wood, with same 1 unit volume as L1.

(L1 and L2 could be cedar, for example, while H1 is ash.)

I am using “density” of the L-wood and H-wood in terms of their weight-mass divided by volume, and I have tables that tell me the densities (or the so-called “specific gravities” of various woods). Assume both L-wood and H-wood are less dense than water.

Assume further that L1 floats on water at its center-line—i.e., such that exactly one hemisphere is above the water and one hemisphere is below.

The answers to the following experimental situations and questions, properly translated into English, should clear up the confusion among the non-physicist paddlers and paddle-makers.

1. Float L1 next to H1.

— Will H1 float above, below or (like L1) at its center-line? It seems intuitive that the denser-hence-heavier H1 will sink down further.

— Which sphere has the greater “buoyant force”, technically defined, pushing it upward as it floats?

2. Float L1 next to L2.

— Will L2 float above, below or (like L1) at its center-line?

— Which sphere has the greater “buoyant force” pushing it upward as it floats?

3. Tether L1, L2 and H1 underwater by a string affixed to the bottom. All three spheres are now floating underwater but none can float upward because of the restraining strings. (The string force is analogous to the paddler force holding a paddle blade under the water.)

— What are the relative upward “buoyant forces” on each of the three spheres, qualitatively or quantitatively? (I'm trying here to understand whether "buoyant force" is different when the object is restrained under water compared to when it is floating at the surface.)

4. Simultaneously cut the strings on L1 and H1.

— Which sphere reaches the water surface first, to its natural float level, or do they both arrive at the same time?

— Which sphere has the greater “buoyant force” acting on it during its upward float journey, between the time the string is cut and the time it settles into its float level? (I'm here trying to understand whether buoyant force on a free-floating underwater object is different from that that on a tethered underwater floating object.)

— Whether the two spheres reach the surface at the same or different times, what, qualitatively or quantitatively, are their relative momentums at the time they reach the surface?

5. Simultaneously cut the strings on L1 and L2.

— Same three questions as in Experiment 4, above.

6. Extra question.

— When the underwater spheres are released and float upward, do they constantly accelerate before they reach the surface? Accelerate and then reach a constant velocity? Accelerate and then decelerate? Something else?

Thanks for your time.

 

[mgb_phys]

You have to be a little careful with the terminology here.
Buoyancy or Buoyant force is just the force pushing up - so only depends on the volume of water displaced.
So a 1m^3 rock, all though it sinks, has a greater buoyancy force than a 0.1m^3 bouy ball that floats.

To know if something floats or not you need the buoyant force up (only depends on the size) and the gravity force down (depends on the mass = size and density) - this is also called the apparent weight or if it floats - the 'flotation'. Thats what's printed on your PFD when it says it has 20lbs of flotation.

So the only thing that decides whether a paddle will float or not, and how much force it is pushed up (or down)by is the density.
If the material is denser than sea water then the bigger you make it the more flotation it will have - note as you make it bigger it will get heavier but it can still float more.
A supertanker is a lot heavier than your kayak but has a lot more flotation.

In practice a blade that is too large will be a hassle for other reasons so you probably want to make a blade from the lightest material possible so that it will have lots of flotation whatever size it is. The limit is going to be how stiff it needs to be to push against the water.