Section 3 - Buoyancy
What is Buoyancy? It is a fundamental principle affecting objects submerged in fluids and was discovered by the Greek
mathematician Archimedes.
Archimedes’s Principle
Any objectcompletely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the
weight of the fluid displaced by the body
Buoyancy – a measure of the upward force a fluid exerts on an object that is submerged/have the ability to float in a
liquid
Buoyant force – the upward force that acts on a submerged object
Buoyant force exist because of pressure differences in fluids
The strength of the buoyant force on an object in water depends on the volume of the object that is under water.
Buoyant force = weight of the fluid displaced by the object
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Let’s look at an example. Below is a rock with a mass of 3 kg and a volume of 1000 cm . Using a balance scale the
weight of the rock is measured to be 29,4 N while the rock is hanging in a stationary position. When the rock is placed in
water however, we see a difference on the balance scale. Now the weight of the rock is measured as 19,6 N. This is called
the apparent weight. The reason for this decrease in weight is the buoyant force that is present as an upwards force
when the rock is placed in water.
True weight = 29,4 N
Apparent weight = 19,6 N
Difference between true weight and apparent weight = BUOYANT FORCE = 29,4 – 19,6 = 9,8 N
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Remember that the volume of the rock is 1000 cm . When the rock is placed in the water, the rock displaces 1000 cm of water.
This is 1 kg. The weight of 1 kg = 9,8 N!
Thus the Buoyant Force = weight of displaced liquid
Buoyancy explains why some objects sink and others float. A submerged object floats to the surface is the buoyant force is
greater than the weight of the object. If the buoyant force is less than an object’s weight, it will sink. We also have to keep in
mind the density of an object. The more dense and object the greater the buoyant force needs to be to keep it afloat. A floating
object displaces just enough water to make the buoyant force equal to the object’s weight. If an object’s density is greater
than the fluid it is placed in, it will sink. If an object’s density is less than the fluid it is placed in, it will float.
Let us now look at the derivation of the formula for the buoyant force.
The pressure at the bottom of the cylinder of water in the container is greater
than the pressure at the top because pressure in a fluid depends on depth.
Fbuoyant = Fbottom – Ftop
(but pressure = F/area and thus F = pressure x area
Fbuoyant = (pressurebottom x area) – (pressuretop x area) (areabottom =areatop)
Fbuoyant = (pressurebottom - pressuretop) x area
pressure in a fluid = ρfgh (where ρf = density of the fluid)
Fbuoyant = (ρfghbottom - ρfghtop) x area
Fbuoyant = ρfg(hbottom - htop) x area
Fbuoyant = ρfgV
(hbottom - htop) x area = volume
(m = ρV)
Fbuoyant = mg = weightfluid
Example 1
A bargain hunter buys a “gold” crown at a flea market. After she gets home, she hangs it from a scale and finds its weight
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to be 7,84 N. She then weighs the crown while it is immersed in water of density 1000 kg/m and then the scale reading was
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6,86 N. Is the crown made of gold? (gold density = 19,3 x 10 kg/m )
Solution 1
buoyant force = true weight – apparent weight = 7,84 – 6,86 = 0,98 N
buoyant force = weight of displaced liquid = wf = ρfgVf (ρf = density of displaced fluid, Vf = volume of displaced fluid)
= 0,98 N
Remember that the object displaces the fluid, thus the volume of the fluid that is displaced is equal to the volume of the object
Vf = Vo = 0,98 / (ρfg) = 0,98 / (1000 x 9,81) = 1 x 10 m
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So the density of the object is ρo = mo / Vo
To get the mass of the object we have to use the true weight: m = w / 9,81 = 7,84 / 9,81 = 0,799184505 kg
ρo = mo / Vo = 0,799184505 / (1 x 10 )7991,845 kg/m
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this is not the density of gold, thus the crown is a fake
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Example 2
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A raft is made of wood having a density of 600 kg/m . The surface area of the raft is 5,7 m and its volume is 0,6 m . When the
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raft is placed in fresh water of density of 1000 kg/m , how much of the raft is below the water level?
Solution 2
“A floating object displaces just enough water to make the buoyant force equal to the object’s weight”
In addition the theory tells us that the buoyant force = weight of displaced fluid
W raft = mraftg
(using the density formula)
wfart = ρraftVraftg
= (600 x 9,81 x 0,6)
= 3531,6 N
In this case then since the object is floating: weight of the displaced fluid(water) = weight of the object = buoyant force
ρwaterVwaterg = ρraftVraftg = B
since Vwater = hwater x areawater
we can use this to find the depth h to which the raft sinks into the water
B = ρwaterVwaterg = ρwater(hwater x areawater)g
then hwater = B / (ρwaterareawaterg
= 3531,6 / (1000 x 5,7 x 9,81)
= 0,063 m