Buoyancy and Archimedes' Principle

      
Introduction

When Andrew tries to lift Jessica out of the water, he observes that Jessica appears to get heavier the farther she gets out of the water.  This illustrates the effect of Buoyant Force.

The Buoyant Force occurs because the pressure in a fluid increases with depth.

P₁  =  _fluidgh₁  and   F₁ =  _fluidgh₁A   Downward force on top of cylinder

P₂  =  _fluidgh₂  and   F₂ =  _fluidgh₂A   Upward force on bottom of cylinder

Buoyant Force =  F₂  -  F₁  =  _fluid ghA  =  _fluid gV

As shown in the diagram above the crown has a mass of 14.7 kg when measured above water and 13.4 kg when measured in water.  Is the crown made of gold?

The apparent weight of the submerged crown, w', equals the actual weight, w, minus the buoyant force, F_B.

w'  =  w   -  F_B  =  w  -  _Fluid gV        thus   w  -  w'  =  _Fluid gV

w  =  _object gV 

  
    w             _object gV         _object               14.7 kg
----------  =  ---------------  =  -----------  =  ---------------------------  =  11.3
w  -  w'        _Fluid gV          _Fluid           14.7 kg  -  13.4 kg

  
Since our fluid was water it is also the specific gravity of the object.  It's density is 11.3 x 10 ³ kg/m² .

Physics Charts - Densities of Substances

 The density charts show gold with a density of 19.3 x 10 ³ kg/m² while lead has a density of 11.3 x 10 ³ kg/m².

The log to the left has a specific gravity of 0.60 and a volume of 2.0 m³.

This gives it a mass of 1200 kg.  

The mass of water displaced is 2000 kg.

Since its buoyant force is ( 2000 kg)g, it will float to the surface.

V _{displaced         object} 
--------------  =  ----------                   This is the relationship for objects that float. 
V _{object             fluid}  

    
A hydrometer measures the specific gravity by how deeply it sinks in a liquid.  The hydrometer to the left consists of a glass tube weighted at the bottom.  It is 25.0 cm long and 2.00 cm² in cross-sectional area with a mass of 45.0 g.  How far from the end should the 1.000 mark be placed in water?

First find the density of the hydrometer.

   
         m                45.0 g
  =  ----  =  ---------------------------  =  0.900 g/cm³    
         V         25.0 cm x 2.00 cm²  

   
V _{displaced         object}           0.900 g/cm³   
--------------  =  ----------     =  ------------------  =  0.900
V _{object             fluid}             1.00 g/cm³   
   

0.900 of the object should be below water.

0.900 x 25.0 cm  =  22.5 cm below water.

Table of Contents

Problems

Table of Contents

Answers

1.	A 30.0 cm hydrometer with a mass of 50.0 g and a cross-sectional area of 2.00 cm2.  When placed in an unknown liquid, it sinks to a depth of 23.5 cm.  What is the density of the unknown liquid?
	m                50.0 g hydrometer  =  ----  =  ---------------------------  =  0.833 g/cm3                            V         30.0 cm x 2.00 cm2      h displaced            hydrometer          ----------------  =  ----------------               Solve for fluid. h hydrometer             fluid                                     object  x h object             0.833 g/cm3 x 30.0 cm   fluid  =  --------------------------  =  ---------------------------------  =  1.06 g/cm3                          h displaced                               23.5 cm
2.	A 5.00 kg copper ball has an apparent mass of 4.35 kg when submerged in an unknown liquid.  What is the density of the unknown liquid?    Physics Charts - Densities of Substances
	w                copper            ----------    =  -----------             Solve for the density of the fluid. w  -  w'           Fluid                              copper    ( w  -  w' )           8.9 g/cm3  x ( 5.00 kg  -  4.35 kg )  Fluid  =  ---------------------------  =  -----------------------------------------------  =  1.16 g/cm3                                  w                                             5.00 kg
3.	A 0.50 kg piece of wood floats in water but is found to sink in alcohol (specific gravity = 0.79) in which it has an apparent mass of 0.035 kg.  What is the specific gravity of the wood?
	w                wood            ----------    =  -------------             Solve for the density of the wood. w  -  w'           alcohol                              alcohol   x  w          0.79  x  0.50 kg   wood  =  ---------------------  =  ----------------------------  =  0.85                         w  -  w'                0.50 kg  -  0.035 kg
4.	The specific gravity of ice is 0.917, whereas that of seawater is 1.025.  What fraction of an iceberg is above the surface of the water?
	V displaced            iceberg          0.917 ----------------  =  ---------------  =  ---------  =  0.895             This is the fraction underwater. V iceberg              seawater         1.025     Fraction above water  =  1.000  -  0.895  =  0.105
5.	A 2.06 kg piece of wood (SG = 0.50) floats on water.  What minimum mass of lead, hung from it by a string will cause it to sink?    Physics Charts - Densities of Substances
	When density is above 1.00, the wood and lead will sink.           m                 m wood  +  m lead     =  ----  =  -----------------------------------  =  1.00             Solve for m lead.          V          V wood  +  m lead / lead       V wood  +  m lead / lead   =   m wood  +  m lead   V wood  -  m wood  =   m lead   -   m lead / lead    =  m lead ( 1  -  lead -1 )                    V wood  -  m wood          4060 cm3  -  2060 g m lead  =  -----------------------  =  -------------------------------  =  2260 g lead                        1  -  lead -1                1  -  ( 11.3 g/cm3 ) -1

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