ICSE Class 9 Physics Chapter 05 Upthrust in Fluids Archimedes Principle and Floatation

Upthrust In Fluids, Archimedes' Principle And Floatation

Syllabus

Buoyancy, Archimedes' principle, floatation, relationship with density; relative density, determination of relative density of a solid.

Scope

Buoyancy, upthrust (Fb); definition; different cases, Fb > = or < weight W of the body immersed; characteristic properties of upthrust; Archimedes' principle; explanation of cases where bodies with density ρ > = or < the density ρ' of the fluid in which it is immersed. R.D. and Archimedes' principle; Experimental determinations of R.D. of a solid and liquid denser than water. Floatation; principle of floatation; relation between the density of a floating body, density of the liquid in which it is floating and the fraction of volume of the body immersed; (ρs/ρL = Vs/V); apparent weight of floating object; application to ship, submarine, iceberg, balloons, etc. Simple numerical problems involving Archimedes' principle and floatation.

Upthrust And Archimedes' Principle

Buoyancy And Upthrust

When a body is partially or wholly immersed in a liquid, an upward force acts on it. This upward force is known as upthrust or buoyant force. It is denoted by the symbol Fb. Thus

The upward force exerted on a body by the fluid in which it is submerged, is called the upthrust or buoyant force.

The property of liquid to exert an upward force on a body immersed in it, is called buoyancy. This property can be demonstrated by the following experiments.

Exp. 1. Pushing An Empty Can Into Water

Take an empty can. Close its mouth with an airtight stopper. Put it in a tub filled with water. It floats with a large portion of it above the surface of water and only a small portion of it below the surface of water.

If we push the can into water, we feel an upward force which opposes the push and we find it difficult to push the can further into water. It is also noticed that as the can is pushed more and more into water, more and more force is needed to push the can further into water, till it is completely immersed. When the can is fully inside water, a constant force is still needed to keep it stationary in that position. Now if the can is released at this position, it is noticed that the can bounces back to the surface and starts floating again.

Exp. 2. Floating A Cork Into Water

If a piece of cork is placed on the surface of water in a tub, it floats with nearly 2/5th of its volume inside water. If the cork is pushed into water and then released, it again comes to the surface of water and floats. If the cork is kept

immersed, our fingers experience some upward force. The behaviour of cork is similar to that of the empty can.

Explanation

When the can or cork is put in the tub of water, two forces act on it: (i) its weight (i.e., the force due to gravity) W which pulls it downwards, and (ii) the upthrust Fb due to water which pushes the can or cork upwards. It floats in the position when the two forces become equal in magnitude (i.e., W = Fb). Now as the can or cork is pushed more and more inside water, the upthrust Fb exerted by water on it increases and becomes maximum (= Fb') when it is completely immersed in water. So when it is released, the upthrust Fb exerted by water on it being greater than its weight W (or force due to gravity), it rises up. To keep the can or cork immersed, an external downward force (= Fb' - W) is needed to balance the net upward force.

Note: Like liquids, gases also have the property of buoyancy, i.e., a body immersed (or placed) in a gas also experiences an upthrust. All objects including ourselves, are also acted upon by a buoyant force due to air, but we do not feel it because it is negligibly small as compared to our own weight. On the other hand, a balloon filled with hydrogen (or any gas less denser than air) rises up because the upthrust (or buoyant force) on balloon due to the surrounding air is more than the weight of balloon filled with the gas.

Condition For A Body To Float Or Sink In A Fluid

When a body is immersed in a fluid, two forces act on the body: (i) the weight W of

the body which acts vertically downwards and (ii) the upthrust Fb which acts vertically upwards. We have noticed that the upthrust depends on the submerged portion of the body. It increases as the submerged portion of body inside the fluid increases and becomes maximum (= Fb') when the body is completely immersed inside the fluid. Fig. 5.1 shows a body held completely immersed in a fluid with two forces W and Fb' acting on it.

Depending upon the density of the fluid, the maximum buoyant force Fb' 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 Fb' (buoyant force when the body is fully immersed).

(i) If Fb' > W or Fb' = W, the body will float (it will not sink). If Fb' > W, the body will float partly immersed with only that much part of it inside liquid, the upthrust Fb due to which becomes equal to the weight W of body (i.e., Fb = W). But if Fb' = W, the body will float with whole of it immersed inside the liquid. Thus for a floating body, net force acting downwards (i.e., apparent weight) is zero.

(ii) If Fb' < W, the body will sink due to the net force (W - Fb') acting on the body downwards. If m is the mass of body, it will go down into the liquid with an acceleration a such that ma = W - Fb' or a = (W - Fb')/m. Here we have ignored the viscous force of the liquid.

Unit Of Upthrust

The upthrust, being a force, is measured in newton (N) or kgf.

Characteristic Properties Of Upthrust

The upthrust has the following three characteristic properties:

(i) Larger the volume of body submerged in a fluid, greater is the upthrust.

(ii) For same volume inside the fluid more the density of fluid, greater is the upthrust.

(iii) The upthrust acts on the body in upward direction at the centre of buoyancy i.e., the centre of gravity of the displaced fluid.

(i) Larger The Volume Of Body Submerged In A Fluid, Greater Is The Upthrust

In the experiment of pushing an empty can or cork into water as described above, it is experienced that the upthrust on the body due to water increases as more and more volume of it is immersed into water, till it is completely immersed.

Similarly, when a bunch of feathers and a pebble of same mass are allowed to fall in air, the pebble falls faster than the bunch of feathers. The reason is that upthrust due to air on pebble is less than that on the bunch of feathers because the volume of pebble is less than that of the bunch of feathers of same mass. However in vacuum, both the bunch of feathers and pebble will fall together because there will be no upthrust.

(ii) For Same Volume Inside The Fluid More The Density Of Fluid, Greater Is The Upthrust

If we place a piece of cork A into water and another identical piece of cork B into glycerine (or mercury), we notice that 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 (or mercury) 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 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 exerts a greater upthrust.

(iii) The Upthrust Acts On The Body In Upward Direction At The Centre Of Buoyancy (i.e., The Centre Of Gravity Of The Displaced Liquid)

For a uniform body completely immersed inside a liquid, the centre of buoyancy coincides with the centre of gravity of the body (Fig. 5.1). But if a body floats in a liquid with its part submerged (Fig. 5.2), the centre of buoyancy B is at the centre of gravity of the displaced liquid (i.e., at the centre of gravity of the immersed part

of the body) which lies below the centre of gravity G of the entire body. The weight of the body W acts downwards at G, while upthrust Fb acts upwards at B such that W = Fb.

Reason For Upthrust

We have read that a liquid contained in a vessel exerts pressure at all points and in all directions. The pressure at a point in liquid is same in all directions (upwards, downwards and sideways). It increases with depth inside the liquid. When a body, say a block of area of cross section A, is immersed in a liquid (Fig. 5.3), the pressure P2 exerted upwards on the lower face of block (which is at a greater depth) is more than the pressure P1 exerted downwards on the upper face of block (which is at a lesser depth). Thus there is a difference in pressure (= P2 - P1) between the lower and upper faces of block. Since force = pressure x area, the difference in pressures due to liquid on the two faces of block causes a net upward force (i.e., upthrust) = (P2 - P1)A on the body. However, the thrust on the side walls of body get neutralised as they are equal in magnitude and opposite in directions.

Note

If a lamina (thin sheet) is immersed in a liquid, the pressure on its both surfaces will be nearly same, so the liquid will exert negligible upthrust on it, causing it to sink into the liquid due to its own weight.

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

Proof: Consider a cylindrical body PQRS of cross-sectional area A immersed in a liquid of density ρ as shown in Fig. 5.4. Let the upper surface PQ of body be at a depth h1 while its lower surface RS be at a depth h2 below the free surface of liquid.

At depth h1, pressure on the upper surface PQ

P1 = h1ρg

Therefore, Downward thrust on the upper surface PQ

F1 = pressure x area = h1ρg A .....(i)

At depth h2, pressure on the lower surface RS

P2 = h2ρg

Therefore, Upward thrust on the lower surface RS

F2 = h2ρg A .....(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 eqns. (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

Fb = F2 - F1

= h2ρgA - h1ρgA

= A(h2 - h1) ρg

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

Therefore, Upthrust Fb = Vρg .....(5.1)

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

Vρg = Volume of solid immersed x density of liquid x acceleration due to gravity.

or Vρg = Volume of liquid displaced x density of liquid x acceleration due to gravity.

= mass of liquid displaced x 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. .....(5.2)

Note

(1) If the body is completely immersed in a liquid, the volume of liquid displaced will be equal to its own volume and upthrust then will be maximum (= Fb').

(2) Although the above result is derived for a cylindrical body, but it is equally true for a body of any shape and size.

Factors Affecting The Upthrust

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

(i) volume of the body submerged in liquid (or fluid), and

(ii) density of the liquid (or fluid) in which the body is submerged.

Effect Of Upthrust

The effect of upthrust is that the weight of body immersed in a liquid appears to be less than its actual weight. This can be demonstrated by the following experiment.

Experiment: Lifting Of A Bucket Full Of Water From A Well

Take an empty bucket and tie a long rope to it. If the bucket is immersed in water of a well keeping one end of rope in hand and the bucket is pulled when it is deep inside water, we notice that it is easy to pull the bucket as long as it is inside water, but as soon as it starts coming out of the water surface, it appears to become heavy and now more force is needed to lift it.

This experiment shows that the bucket of water appears lighter when it is immersed in water than its actual weight (in air).

Similarly, when pulling a fish out of water, it appears lighter inside water as compared to when it is out of water.

Similarly, a body weighed by a sensitive spring balance, will weigh slightly less in air than in vacuum due to upthrust of air on the body.

Archimedes' Principle

When a body is immersed in a liquid, it occupies the space which was earlier occupied by the liquid i.e., it displaces the liquid. The volume of liquid displaced by the body is equal to the volume of the submerged part of the body so the body experiences an upthrust equal to the weight of the liquid displaced by it.

It is the upthrust due to which a body immersed in a liquid appears to be of weight less than its real weight. The apparent loss in weight is equal to the upthrust on the body. This is called the Archimedes' principle. Thus

Archimedes' principle states that when a body is immersed partially or completely in a liquid, it experiences an upthrust, which is equal to the weight of the liquid displaced by it.

This principle applies not only to liquids, but it applies equally well to gases also.

Experimental Verification Of Archimedes' Principle

Archimedes' principle can be verified by either of the following experiments.

Expt. (1)

Take two cylinders A and B of the same volume. The cylinder A is solid and the cylinder B is hollow. Suspend the two cylinders from the left arm of a physical balance keeping the solid cylinder A below the hollow cylinder B. Then balance the beam by keeping weights on right arm of the balance. In this situation, both cylinders A and B are in air.

The solid cylinder A is now completely immersed into water contained in a beaker D placed on a bench C as shown in Fig. 5.5, taking care that the cylinder A does not touch the sides and bottom of the beaker. It is observed that the solid cylinder A loses weight i.e., the left arm of the balance rises up. Obviously the loss in weight is due to upthrust (or buoyant force) of water on the cylinder A.

Now pour water gently in the hollow cylinder B till it is completely filled. It is observed that the beam balances again.

Thus, it is clear that the buoyant force acting on solid cylinder A is equal to the weight of water displaced by it. Since the cylinders A and B both have equal volume, so the weight of water in the hollow cylinder B is just equal to the weight of water displaced by the cylinder A. Hence the buoyant force acting on the cylinder A is equal to the weight of water displaced by it. Thus, it verifies the Archimedes' principle.

Expt. (2)

Take a solid (say, a metallic piece). Suspend it by a thin thread from the hook of a spring balance [Fig. 5.6(a)]. Note its weight.

Now take a eureka can and fill it with water up to its spout. Arrange a measuring cylinder below the spout of the eureka can.

Now immerse the solid gently into water of the eureka can. The water displaced by it gets collected in the measuring cylinder [Fig. 5.6 (b)]. When water stops dripping through the spout, note the weight of the solid and the volume of water collected in the measuring cylinder.

In Fig. 5.6, the solid weighs 300 gf in air and 200 gf when it is completely immersed in water. The volume of water collected in the measuring cylinder is 100 ml i.e., 100 cm3.

Therefore, Loss in weight = 300 gf - 200 gf = 100 gf .....(i)

Volume of water displaced = Volume of solid = 100 cm3

Since density of water = 1 g cm-3

Therefore, Weight of water displaced = 100 gf .....(ii)

From eqns. (i) and (ii)

Weight of water displaced = Upthrust or loss in weight.

Thus the weight of water displaced by solid is equal to the loss in weight of the solid. This verifies Archimedes' principle.

Teacher's Note

When you take a bath, you might notice how your body feels lighter when submerged in water - this is the upthrust in action, which is why swimming feels effortless compared to moving on land.

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