Upthrust in fluids: Buoyancy, experiment, conditions required, characteristic properties, reason for upthrust and mathematical proof.

UPTHRUST IN FLUIDS

Upthrust

The upward force exerted on a body by the fluid in which it is submerged, is called the upthrust  force in liquids or buoyant.Upthrust in liquid  is denoted by the symbol F_{B .}                         
The property of liquid to exert an upward force on a body immersed in it,is called buoyancy.

EXPERIMENT

Need:

• A empty can with a air tight stopper on its open end.
• A container filled with water.

An empty can on water

Working:

• Put the can over  the water

Observation:

• It can be seen that the can with its larger portion floats over the water and a smaller portion of is inside the water.
• On pushing the can inside the water it tries to come out because of the upthrust on it.
• It is also noticed that more and more the can is pushed greater the upthrust becomes.
• When the can is completely inside the water a constant force is required to keep that in its place. And it is the maximum upthrust on the can applied.

CONDITIONS FOR A BODY TO FLOAT OR SINK IN A FLUID

When a body is immersed in a liquid two forces acts on it:

• Weight W (acting vertically downward).
• Upthrust  F_B   (acting upward).

Forces acting on a body immersed inside  a liquid

It is noticed that the upthrust depends on the submerged portion of the body.It increases with the increase in the submerged portion of the body and it becomes maximum(=F’_B ) when the body is completely immersed inside the fluid.
Depending on the density of the fluid,the maximum buoyant force F’_B can be greater than,equal to or less than the weight W of the given body.Whether the body will float or sink in a fluid, depends on the relative magnitudes of forces W and F’_B( buoyant force when the body is fully immersed).

CHARACTERISTIC PROPERTIES OF UPTHRUST

There are three characteristic properties of upthrust.They are follows:-

1.Larger the volume of the body submerged in a fluid, greater is the upthrust: In the before discussed experiments it could be seen that the upthrust increases with an increase in volume of the object in the fluid.Similarly,when a bunch of feathers and a pebble of some mass are allowed to fall in air, the pebble falls faster than the bunch  of feathers. The reason is that on the bunch of feathers because the volume of pebbles is less than that of the bunch of feathers of same mass.However in vacuum,both the bunch of feather and pebble will fall together because there will be no upthrust.

2.For some volume inside the fluid more the density of the liquid greater is the upthrust: If we keep a piece of cork A in water and a piece of cork B in glycerine (or mercury) we notice that both A and B floats but the volume of cork B immersed in glycerine (or mercury) is smaller as compared  to the volume of cork A immersed in water.The reason is that the density of glycerine is more than that of water.Now if we want to immerse  cork B in glycerine to the same extent  as cork A in water,then an additional force is needed on the cork B,to immerse it to the same level as cork A.This shows that for same volume of a body inside the liquid,a denser liquid , a denser liquid exerts a greater upthrust.

3.The upthrust acts on the body in upward direction at the center of buoyancy(i.e.,the center of gravity of the displaced liquid): For a uniform body completely immersed inside a liquid,the center of buoyancy coincides with the center of gravity of the body.But if the body floats in a liquid with with its part submerged,the center of buoyancy B is at the center of gravity of the displaced liquid which lies below the center of the gravity G of the entire body.The weight of the body W acts downwards at G,while upthrust  F_B acts upon at B such that W = F_B  .


REASON FOR THE UPTHRUST



Reason for the upthrust

We know that a liquid contain in a vessel exerts pressure in all directions.The pressure at a point is same in all directions.It increases with an increase in depth  inside the liquid.When a body, say a block of area of cross section A,is immersed in  a liquid, the pressure P₂ exerts upwards on the lower face of the block(which is at a greater depth) is more than the pressure P₁ exerted downwards on the upper face of the block (which is at a lesser depth). Thus there is a difference in pressure (= P₁ - P₂) between the lower and upper faces of the block.Since force=pressure*area, the difference in pressures due to liquid on the two faces of block causes a net upward force(i.e.,upthrust)=(P₁ - P₂)A on the body.However, the thrust F on the side walls of the body get neutralised as they are equal in magnitude and opposite in direction.

UPTHRUST IS EQUAL TO THE WEIGHT OF DISPLACED LIQUID (Mathematical proof)
When a body is immersed in a liquid, upthrust on it due to liquid is equal to the weight of the liquid displaced by the submerged part of the body.

Upthrust on different volume object

Proof:

Consider a cylinder body PQRS of cross-section area A immersed in a liquid  of density.Let the upper surface PQ of the body be at at a depth  h₁  while its lower surface RS be t a depth h­₂ below the free surface of liquid.

At depth h₁ , pressure on the upper surface PQ 

 P₁  =  h₁ƿg

Therefore,downward thrust on the upper surface PQ   

F₁ = pressure*area = h1ƿgA...(i)

At depth h­₂, pressure on the lower surface RS 

P₂ = h­₂ ƿg

Therefore, upper thrust on the lower surface RS   

F₂ = h­₂ƿgA...(ii)

The horizontal thrust at various points on the vertical sides of body get balanced because liquid pressure is same at all points at the same depth.

From above equations (i) and (ii), it is clear that F₂  >F₁ because h­₂ > h₁ and therefore,the body will experience a net upward force.

Resultant upward thrust (or buoyant force) on the body

                                  F_B  = F₂ - F₁  

                                       = h­₂ƿgA - h₁ƿgA

                                       =A( h­₂ - h₁ )ƿg

But A( h­₂ - h₁ ) = V, the volume of the body submerged in liquid.

Therefore,

Upthrust F_B  = Vƿg

Vƿg = Volume of the solid immersed * acceleration due to gravity.

Since a solid when immersed in a liquid, displaces liquid equal to the volume of its submerged part, therefore

Vƿg = Volume of liquid displaced * density of liquid * acceleration due to gravity.

    = mass of liquid displaced * acceleration due to gravity.

    = Weight of the liquid displaced by the submerged part of the body.

Hence,

      Upthrust = Weight of the liquid displaced by the submerged part of the body.

FACTORS AFFECTING THE UPTHRUST
From the above discussion, it is clear that the magnitude of upthrust on the  body due to a liquid (or fluid) depends on the following two factors:

• volume of the body submerged in liquid and
• density of the liquid in which the body is submerged.