# Sect. 10-7: Buoyancy/Archimedes Principle

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24-Feb-2016Category

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### Transcript of Sect. 10-7: Buoyancy/Archimedes Principle

Sect. 10-7: Buoyancy/Archimedes PrincipleExperimental facts: Objects submerged (or partially submerged) in a fluid APPEAR to weigh less than in air.When placed in a fluid, many objects float! Both are examples of BUOYANCY.

Buoyant Force: Occurs because the pressure in a fluid increases with depth!P = g h (fluid at REST!!)

Archimedes Principle The total (upward) buoyant force FB on an object of volume V completely or partially submerged in a fluid with density F: FB = FVg (1) FV mF Mass of fluid which would take up same volume as object, if object were not there. (Mass of fluid that used to be where object is!) Upward buoyant force FB = mFg (2)FB = weight of fluid displaced by the object!(1) or (2) Archimedes Principle Proved for cylinder. Can show valid for any shape

Archimedes Principle

Object, mass m in a fluid. Vertical forces are buoyant force, FB & weight, W = mgApparent weight = net downward force:

W Fy = W - FB < W

Object appears lighter!

Archimedes Principle & Bath Legend

Archimedes Principle: Valid for floating objectsFB = mFg = FVdispl g (mF = mass of fluid displaced, Vdispl = volume displaced)W = mOg = OVOg (mO = mass of object, VO = volume of object)Equilibrium: Fy = 0 = FB - W

Archimedes Principle: Floating objectsEquilibrium: Fy = 0 = FB -W FB = W or FVdispl g = OVOg

f = (Vdispl/V) = (O/F) (1)f Fraction of volume of floating object which is submerged.Note: If fluid is water, right side of (1) is specific gravity of object!

Example: Floating log

(a) Fully submerged: FB > W Fy = FB -W = ma (It moves up!)(b) Floating: FB = W or FVg = OVg Fy = FB -W = 0 (Equilibrium: It floats!)

Prob. 33: Floating Iceberg!(SG)ice= 0.917 (ice/water), (SG)sw= 1.025 (sw/water)What fraction fa of iceberg is ABOVE waters surface? Iceberg volume VOVolume submerged VdisplVolume visible V = VO - Vdispl Archimedes: FB = swVdisplgmiceg = iceVOg Fy= 0 = FB - miceg swVdispl = iceVO (Vdispl/VO)= (ice/sw) = [(SG)ice/(SG)sw] = 0.917/1.025 = 0.89fa = (V/VO) = 1 - (Vdispl/VO) = 0.11 (11%!)

Example 10-9: Hyrdometer

(O/F)= (Vdispl/V)

Prob. 22: Moon Rock in water Moon rock mass mr = 9.28 kg. Volume V is unknown. Weight W = mrg = 90.9 NPut rock in water & find apparent weight W = mag apparent mass ma = 6.18 kg W = 60.56 N. Density of rock = (mr/V) = ?

W Fy = W - FB = mag . FB = Buoyant force on rock.Archimedes: FB = waterVg. Combine (g cancels out!):mr - waterV = ma . Algebra & use definition of : V = (mr - ma)/water. = (mr/V) = 2.99 103 kg/m3

Example 10-10:Helium BalloonAir is a fluid Buoyant force on objects in it. Some float in air.What volume V of He is neededto lift a load of m=180 kg? Fy=0 FB = WHe + WloadFB = (mHe + m)g , Note: mHe = HeVArchimedes: FB = airVg airVg = (HeV + m)g V = m/( air - He) Table: air = 1.29 kg/m3 , He = 0.18 kg/m3 V = 160 m3

Prob. 25: (Variation on example 10-10)Spherical He balloon. r = 7.35 m. V = (4r3/3) = 1663 m3mballoon = 930 kg. What cargo mass mcargo can balloon lift? Fy= 0 0 = Fbouy - mHeg - mballoon g - mcargogArchimedes: Fbouy = airVgAlso: mHe = HeV, air = 1.29 kg/m3, He = 0.179 kg/m3

0 = airV - HeV - mballoon - mcargo

mcargo = 918 kg

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