Flotation, Archimedes’ Principle, Law of flotation, Practice problems, FAQs

You may have seen the childrens playing with the boat during the rain. Usually these boats are made of wood or paper. When these boats are placed in the water, it moves with the flow of water. When the boat fills with the rain water it gets sinked in the water. Do you know why it is floating initially and sinks when it is filled with water? Initially the density of the boat was less than the density of the water as it is made of wood. So the boat is floating. When the boat is filled with the rain water its density becomes equal to the density of the water consequently it submerges in the water. In the page we will learn about floating and various concepts relating to it.

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Table of content

Flotation
Buoyancy and buoyant force
Archimedes’ Principle
Law of flotation
Practice problems
FAQs

Flotation

The term "flotation or floating" refers to an object's tendency to rise to the top of a fluid or to float on its surface either completely or partially. If the object is not floating and going down to the bottom of the liquid then it is said to be sinking. The floating or sinking depends on the densities of the object to be floated and the fluid in which it is floating. If the density of the object is less than the density of the fluid then the object will be said to be floating. On the other hand if the density of the object is greater than the density of fluid then the object is said to be sinking.

To understand the flotation let us consider an example. Take two blocks, one is of wood and other of the aluminium. If we dip these blocks in the water, then the block of wood will float on the surface of the water and the aluminium will sink in the water as shown in the above figure. This is due to the fact that the density of the wood is less than the density of the water and the density of the aluminium is greater than the density of water.

Buoyancy and buoyant force

If an object is fully or partially submerged in a fluid, the object experiences an upward push by the fluid. This phenomenon of experiencing the push is called buoyancy. And the push is called buoyant force.

To observe the buoyancy and buoyant force take a ball of cork and submerge it into water. It will float on the surface of the water. Now push the cork ball and take it deep into the water. As you release the ball it will come to the surface of the water by itself. There is buoyant force which takes the ball to the surface.

The magnitude of buoyancy force can be calculated by Archimedes’ principle.

Archimedes’ Principle

When a solid object is fully or partially submerged in a fluid, it displaces the water and due to the buoyant force of the fluid, it seems to be lighter than it is in the air.

According to the Archimedes’ Principle, “ The buoyant force is equal to the weight of the fluid which is displaced by the solid object.”

Suppose a solid block of Height H and area A is submerged into the water upto the height x as shown in figure. We know the density of the material is define as,

$D e n s i t y (ρ) = \frac{M a s s (M)}{V o l u m e (V)}$

If density of the solid block is s and volume of the block is Vs=HA then

mass of the block $M_{s} = ρ_{s} \times V_{s}$

So the weight of the block $W_{s} = M_{s} \times g = ρ_{s} \times V_{s} \times g . . . . . (i)$

If density of the fluid is f and volume of the fluid displaced is Vd=xA

then mass of the fluid displaced $M_{f} = ρ_{f} \times V_{d}$

Also the weight of the fluid displaced $W_{f} = M_{f} \times g = ρ_{f} \times V_{d} \times g . . . . . (i i)$

Hence According to the Archimedes’ Principle the magnitude of buoyancy force is

$F_{B} = W_{f}$

$F_{B} = ρ_{f} \times V_{d} \times g$

Where, f= density of the fluid

Vd= Volume of fluid displaced, in this case it is equal to xA.

This is the buoyancy force on the block due to the fluid.

Law of flotation

Depending on the relative magnitudes of the object's weight (W) and the buoyant force (FB) that the liquid exerts. An object submerged in a liquid will either partially float, float, or sink, If

Weight of the object is greater than the buoyant force, (W>FB) then the body will sink in the liquid. It is possible when density of the solid is greater than the density of the liquid (s>L). The apparent weight of the object will be the difference of weight of the object and the buoyancy force.
Weight of the object is equal to the buoyant force, (W=FB) then the body will float in the liquid. It is possible when density of the solid is equal to the density of the liquid (s=L). The apparent weight of the object will be equal to zero as weight of the object is equal to the buoyant force.
Weight of the object is less than the maximum buoyant force (which can be calculated by pushing it completely into the liquid), (W>FB) then the body will partially float such that the buoyancy force balances the weight of the liquid in the liquid. It is possible when density of the solid is less than the density of the fluid (s<L). The apparent weight of the object will be equal to zero as weight of the object is equal to the buoyant force.

Practice problems

Q. A block of wood has the following measurements: length = 5 m, width = 5 m, and height = 6 m. What will the buoyant force on the block be if it is entirely submerged in the water?

A. Given length = 5 m, width = 5 m, and height = 6 m

So the volume of the water displaced $V_{d} = l \times w \times h = 5 \times 5 \times 6 = 150 m^{3}$

Density of the water, f=1000 kg/m3

The buoyant force can be given by,

$F_{B} = ρ_{f} \times V_{d} \times g$

$F_{B} = 1000 \times 150 \times 9.81$

$F_{B} = 1471.5 k N$

Hence the buoyant force is 1471.5 kN.

Q. A object of density 1300 kg/m3 is completely submerged in the water. If the volume of the object is 0.5 m3, find the apparent weight of the object.

A. Given s=1300 kg/m3 and Vs=0.5 m3.

The weight of the object $W_{s} = ρ_{s} \times V_{s} \times g$

The buoyant force $F_{B} = ρ_{f} \times V_{d} \times g$

The apparent weight of the object is the difference of the weight of the object and the buoyant force,

$W_{a p p} = W_{s} - F_{B}$

$W_{a p p} = ρ_{s} \times V_{s} \times g - ρ_{f} \times V_{d} \times g$

As object is completely submerged $V_{s} = V_{d} = V$

$W_{a p p} = (ρ_{s} - ρ_{f}) \times V \times g$

$W_{a p p} = (1300 - 1000) \times 0.5 \times 9.81$

$W_{a p p} = 1.47 k N$

Q. A block of density 800 kg/m3 is submerged into a liquid of the same density. If the volume of the block is 0.1 m3, then what is the buoyant force on the block and apparent weight of the block?

A. Given s=800 kg/m3 and l=800 kg/m3

Vs=0.1 m3

The buoyant force on the block is,

$F_{B} = ρ_{f} \times V_{d} \times g$

As the densities are equal, so it is the condition of floating here $V_{d} = V_{s}$

$F_{B} = ρ_{f} \times V_{s} \times g$

$F_{B} = 800 \times 0.1 \times 9.81$

FB=784.8 N

Now the weight of the block

$W = ρ_{s} \times V_{s} \times g$

$W = 800 \times 0.1 \times 9.81$

$W = 784.8 N$

Hence the buoyancy force will be 784.8 N. The weight of the block will equal to the buoyant force so the apparent weight will be zero.

Q. If a piece of wood with a density of 900 kg/m3 is floating in water with a density of 1000 kg/m3. How much of the wood’s volume is outside the water?

A. Given s=900 kg/m3 and l=1000 kg/m3

It is the condition of partial flotation. Here the buoyant force will balance the weight of the wood.

$W = F_{B}$

$ρ_{s} \times V_{s} \times g = ρ_{f} \times V_{d} \times g$

$900 \times V_{s} = 1000 \times V_{d}$

$V_{d} = 0.9 V_{s}$

The 90 % volume is inside the water. So 10 % is outside the water.

Hence wood’s volume outside the water is 10 %

FAQs

Q. A solid object of density 500 kg/m3 will (float / partially float / sink) in the fluid of density 400 kg/m3 . Explain.
A. As the density of the solid is greater than the fluid. So the buoyant force will be less than the weight of the liquid. Hence it will sink.

Q. What Causes an Airship to Float?
A. Helium and hydrogen gases are used by the airship to float in the sky since they are lighter and aid in the airship's ascent.

Q. What causes a submarine to float and sink?
A. When sink into the water, it fills the weight tanks on either side with water, and when it has to float, the tank is empty. The specialised pumps are used to fill and empty the tanks.

Q. How is buoyancy related to density?
A. The buoyant force, often known as buoyancy, is directly proportional to the fluid's density.