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Deidre Sullivan

Density and Specific Gravity Density: ρ=m/V where ρ is density, m is mass, and V is volume. The density of water is: ρwater = 1 g/ml = 1 g/cm3 = 1 kg/l

Specific gravity is the density of something compared to water. Specific gravity = ρobject/ρwater

The specific gravity of water is 1

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Water Pressure Increases with Depth

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Meters

Buoyancy

Buoyancy is an upward acting force caused by the difference in fluid pressure pushing on the top and bottom of an object.

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Buoyant Force

Fb

Archimedes’ Principle

The magnitude of the buoyant force acting on an object is equal to the weight of the fluid displaced (i.e. pushed out of the way) by the object.

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Buoyant and Gravitational Forces The magnitude of the buoyant force always

equals the weight of the displaced fluid.

𝑭𝑭𝒃𝒃 = ρ𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇g𝑽𝑽𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 = 𝑾𝑾𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇

The gravitation force is weight of the object. 𝑭𝑭𝒈𝒈 = 𝒎𝒎𝒈𝒈 (weight of the object)

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Newtons (N) are a convenient unit to measure these forces. N = ρgVdisplaced 1 N = 1 kg . m/s2, but to make calculations easy we will use the weight of objects measured in grams.

Will an object float or sink? 𝑭𝑭𝒃𝒃 = 𝑾𝑾𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 𝑭𝑭𝒈𝒈 = 𝒎𝒎𝒈𝒈

𝑭𝑭𝒏𝒏𝒏𝒏𝒏𝒏 = 𝑭𝑭𝒈𝒈 − 𝑭𝑭𝒃𝒃

If ρ𝒐𝒐𝒃𝒃𝒐𝒐𝒏𝒏𝒐𝒐𝒏𝒏 < ρ𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 then 𝑭𝑭𝒈𝒈 < 𝑭𝑭𝒃𝒃 and 𝑭𝑭𝒏𝒏𝒏𝒏𝒏𝒏 will be negative. The object will float or will rise in the water column. If ρ𝒐𝒐𝒃𝒃𝒐𝒐𝒏𝒏𝒐𝒐𝒏𝒏 > ρ𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 then 𝑭𝑭𝒈𝒈 > 𝑭𝑭𝑩𝑩 and 𝑭𝑭𝒏𝒏𝒏𝒏𝒏𝒏 will be positive. The object will sink or will fall in the water column.

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𝑭𝑭𝒃𝒃

𝑭𝑭𝒈𝒈 (𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰 𝐨𝐨𝐨𝐨 𝐝𝐝𝐰𝐰𝐝𝐝𝐝𝐝𝐝𝐝𝐝𝐝𝐝𝐝𝐰𝐰𝐝𝐝 𝐨𝐨𝐝𝐝𝐟𝐟𝐰𝐰𝐝𝐝)

(𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰 𝐨𝐨𝐨𝐨 𝐨𝐨𝐨𝐨𝐨𝐨𝐰𝐰𝐝𝐝𝐰𝐰)

Floating Object ρ𝒐𝒐𝒃𝒃𝒐𝒐𝒏𝒏𝒐𝒐𝒏𝒏 𝑽𝑽𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇

ρ𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 V𝒐𝒐𝒃𝒃𝒐𝒐𝒏𝒏𝒐𝒐𝒏𝒏

𝑽𝑽𝒐𝒐𝒃𝒃𝒐𝒐𝒏𝒏𝒐𝒐𝒏𝒏 ≠ 𝑽𝑽𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 𝒇𝒇𝒇𝒇𝒅𝒅𝒅𝒅𝒇𝒇𝒅𝒅𝒐𝒐𝒏𝒏𝒇𝒇

How much does the ducky weigh when it is floating on water?

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=

What if the object doesn’t float or sink?

If ρ𝒐𝒐𝒃𝒃𝒐𝒐𝒏𝒏𝒐𝒐𝒏𝒏 = ρ𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇

the object will neither float nor sink, it is neutrally buoyant

F𝒏𝒏𝒏𝒏𝒏𝒏 = 𝑭𝑭𝒈𝒈− F𝒃𝒃 = 0

𝑭𝑭𝑩𝑩

𝑭𝑭𝒈𝒈

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Apparent Weight or Wet Weight

Reading of apparent weight

in water.

Wwater = Wair- Fb Reading of weight in air.

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Wair

Wwater

Calculating volume of common shapes

V = a3 V = πr2h

V = 4/3πr3 V = abc

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Trick to calculating the volume of odd-shaped objects

Insert float into a beaker of water and measure the amount of water displaced.

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Trick to measuring the buoyant force of odd-shaped objects

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P 266 in chapter 6

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Buoyancy Terms Buoyancy – The upward force exerted on the ROV cause

by the displacement of water. Ballast – Weight added to the ROV to reduce its

buoyancy. Positively Buoyant – Net buoyancy forces are forcing the

ROV up to the surface of the water. Negatively Buoyant – Net buoyancy forces are causing

the ROV to sink. Neutrally Buoyant – Net buoyancy forces cause the ROV

to neither sink or float.

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Buoyancy and Ballast

Buoyancy should be located at the top of the ROV. Ballast should be located at the bottom of the ROV. The greater the separation between the two, the greater

the stability of the ROV. Add enough buoyancy to provide for late arrivals added

to the ROV. Use the ballast to trim the buoyant forces and level the

ROV from front to rear and side to side.

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Is it better to add floats or weights?

Are you better off designing a slightly positive vehicle and adding weights to trim it , or designing a slightly negative vehicle and adding floats?

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Why things in water tip or flip Center of Buoyancy vs. Center of Gravity

The physical distance between the center of buoyancy (CB) and the center of gravity (CG) is called BG. The larger the BG, the more stable the ROV, whether at the surface or underwater.

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Would you ever attach a life jacket to your feet?

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Buoyancy Calculation Attach a scale to the ROV and drop it into the water. Measure the ROV weight with it completely submerged.

Make sure that you give time for all the bubbles to escape.

Record the weight in kilograms In freshwater, 1 kg of weight can be neutralized by 1 liter

of water displacement. 1000 grams can be displaced by 1000 cubic centimeters

of freshwater. This makes our calculations SO EASY.

For move information on Buoyancy, Stability, and Ballast see Underwater Robotics, Chapter 6 20

If we need about 1 kg of buoyancy, what can we use?

Water bottles. 500ml each. One on port and one on starboard. Not glamorous, but will work in the pool. Has a problem with crushing at deeper depths.

Foam – This can be big problem with the wrong density foam. Many ROVs have been built with pipe insulation foam, only to fall and not get back up when they get down to the deep end of the pool. The water pressure crushes the foam and the ROV becomes negatively buoyant.

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Plumbing Pipe Plumbing pipe with the ends capped make great sources

of buoyancy. Will maintain structure and water displacement to the

deepest of pools. Easy to build and put on the ROV. The water easily passes around it unlike a big foam stop

sign.

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PVC or ABS pipe? PVC is negatively buoyant material. ABS is positively buoyant material. 10 feet of 2” PVC weighs 3kg 10 feet of 2” ABS weighs 1.6kg

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Buoyancy of PVC tubes (sealed)

1” PVC 1 ¼” PVC 1 ½” PVC 2” PVC

Diameter (cm) 3.340 4.216 4.826 6.033 Weight in Air

(g/cm) 2.532 6.357 7.174 10.000

Volume per cm (cc) 8.762 13.960 18.292 28.581

Net Buoyancy (g/cm) [weight in

air per cm - Volume per cm]

-6.230 -7.603 -11.118 -18.581

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Weight Statement Table Project: Summer Institute ROV Building Water Density: 1g/cc Last Updated: 7/29/13 Last Update By: you

SubSys Part(s) Part Weight Part Shape Dimensions Volume of

Displaced Water

Weight of Displaced

Water Buoyancy

(Fg) (cc) (Fg) (Fb)

PVC Pipe 393.73 cylinder ᴓ6.027cmx40c

m 1141.17 1141.17 747.44

ABS Pipe 209.96 cylinder ᴓ6.027cmx40c

m 1141.17 1141.17 931.21 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Totals: 603.69 2282.34 2282.34 1678.65

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