Contents

By learning Physics Topics, we can gain a deeper appreciation for the natural world and our place in it.

## What are the Three Conditions for a Floating Body?

Floatation And Immersion Of A Body In A Liquid At Rest

When a body is immersed in a liquid at rest, then the following two forces act on the body simultaneously.

- The weight W
_{1}of the body which acts vertically downwards through the centre of gravity of the body. - The buoyancy of the liquid W
_{2}(weight of the liquid displaced by the body) which acts vertically upwards

through the centre of buoyancy.

The value of W_{1} may be greater than, equal to, or less than W_{2} and hence any one of the three cases may arise.

Case 1 : If W_{1} > W_{2}, i.e., the weight of the body is greater than the weight of the liquid displaced by the body, then the body will sink in the liquid.

In this case, the net downward force = W_{1} – W_{2}

If the body is homogeneous and if its density and volume are ρ_{1} and V_{1} respectively, then W_{1} = V_{1}ρ_{1}g. If the density of the liquid is ρ_{2}, then the weight of the liquid displaced by the body, W_{2} = V_{1}ρ_{2}g.

∴ If W_{1} > W_{2}, then ρ_{1} > ρ_{2}.

So, if the density of the material of the body is greater than the density of the liquid, then the body sinks in the liquid. For example, a piece of stone or iron sinks in water.

Case 2: If W_{1} = W_{2}, i.e., the weight of the body is equal to the weight of the liquid displaced by the body, then the body will float at rest in any position in the liquid and be totally immersed in that liquid.

In this case, the weight of the body and the buoyancy balance each other and hence the net force acting on the body becomes zero. So, the body becomes apparently weightless.

Since, W_{1} = W_{2}, V_{1}ρ_{1}g = V_{1}ρ_{2}g or, ρ_{1} = ρ_{2}.

Hence if the density of the material of the body is equal to the density of the liquid, then the body can float remaining totally immersed within the liquid.

Case 3: If W_{1} < W_{2}, i.e., the weight of the body is less than that of the liquid displaced by it, then the body floats remaining partially immersed in the liquid.

If the body is released after immersing it completely inside the liquid, then under the influence of the force (W_{2} – W_{1}), it will move up through the liquid and, after some time, a part of the body will gradually come out of the liquid. As a result, the amount of liquid displaced by the body is less. When the weight of the body is equal to that of the liquid displaced by it, the body floats remaining partially immersed in the liquid.

Since, W_{1} < W_{2}, ρ_{1} < ρ_{2}.

So, when the density of the material of the body is less than the density of the liquid, the body floats on the liquid. For example, the density of wood is 0.7 – 0.9 g cm^{3} and that of pure water is 1 g ᐧ cm^{-3}.

So, a piece of wood floats on water. Again, as the density of iron (7 – 7.9 g ᐧ cm^{-3}) is less than that of mercury (13.68 g ᐧ cm^{-3}), a piece of iron floats on mercury.

A floating body has no apparent weight: When a body floats in a liquid, the weight of the body becomes equal to the weight of the liquid displaced by the body. According to Archimedes’ principle, the apparent loss of weight of the body inside the liquid is equal to the weight of the liquid displaced by that body, and hence the apparent weight of the body in the liquid = weight of the body – weight of liquid

displaced by the body = 0.

So, a floating body in a liquid has no apparent weight. The weight of the floating body is balanced by the buoyant force exerted by the liquid.

### PROPERTIES OF A FLOATING BODY

Conditions of Equilibrium of a Floating Body

Two equal but opposite forces act on a floating body—one is the weight of the body which acts downwards and another is the buoyancy of the liquid which acts in the upward direction. To keep the floating body in equilibrium, these two forces must act along the same straight line. Otherwise, these two forces will constitute a couple and will produce a rotational motion in the body. Hence, for the equilibrium of a floating body, the following two conditions must be satisfied.

- Condition of floatation: The weight of the floating body must be equal to the weight of the liquid displaced by the body.
- Condition of equilibrium: The centre of gravity of the body and the centre of buoyancy must lie on the same vertical line.

In Fig., a body is floating in equilibrium. The centre of gravity of the body is G, centre of buoyancy is B and weight of the body (W_{1}) = weight of the displaced liquid (W_{2}). In equilibrium, the vertical line ab passes through the centre of gravity G and the centre of buoyancy B. The line ab is called the centre line.

### Stability of a Floating Body

The equilibrium of a floating body may be of three types—

- neutral equilibrium,
- stable equilibrium and
- unstable equilibrium. If the floating body is tilted slightly, then the state of its equilibrium can be studied.

If a slight tilt of the floating body from its equilibrium position does not produce any shift of the centre of buoyancy, i.e., if the new centre of buoyancy coincides with its previous position, then the body is said to be floating in neutral equilibrium. In this case, the centre of gravity of the body and the centre of buoyancy lie on the same vertical line. A sphere floating on a liquid is always in neutral equilibrium [Fig.(a)].

stable equilibrium and unstable equilibrium:

Usually, when a floating body is tilted, the centre of buoyancy shifts towards the leaning side as more liquid is

displaced on that side of the body. Let B_{1} be the new position of the centre of buoyancy [Fig.(b) and (c)]. In this situation, the weight of the body (W_{1}) and the buoyant force (W_{2}) do not act along the same vertical line. These two equal but parallel forces constitute a couple. If this couple can return the body to its original position, then the equilibrium is said to be stable [Fig.(b)]. But if the couple does not return the body to its original position, but rather it tends to tilt the body further, then the equilibrium is said to be unstable [Fig.(c)].

Metacentre: In the tilted position of a floating body, the point at which the vertical line drawn through the centre of buoyancy (B_{1}) cuts the centre line ab is called the metacentre (M) of the body.

From Fig., it is understood that if the metacentre (M) lies above the centre of gravity (G) of the body, then the equilibrium of the body is stable. But if the metacentre (M) lies below the centre of gravity (G) of the body, then the equilibrium is unstable. Hence, for stable equilibrium of a floating body, the metacentre of the body should be situated above its centre of gravity. In the case of neutral equilibrium, the points G and M coincide with each other. The distance GM is called metacentric height. The more the distance of the centre of gravity of the body below the metacentre, the stronger the moment of the restoring couple acting on the body and the equilibrium will be more stable.

Stability of ships and boats: Boats, ships, etc., are so constructed that they can float in water at stable equilibrium. For this, two conditions should be followed.

- The external shapes of these vehicles are so designed that the metacentre lies at a suitable height. For this, the width of the vehicles are made greater at the top than at the bottom.
- The centre of gravity of these vehicles is kept as low as possible by filling the bottom with heavy cargo. For this, some additional weights, called ballast, are kept loaded in the hold of an empty ship.

### Two Important Relations in Connection with a Floating Body

i) Let a body of volume V and density D be allowed to float on a liquid of density d. If the volume of the immersed part of the body in that liquid is v, then the volume of liquid displaced by the body will also be v.

So, according to the condition of floatation,

weight of the body = weight of the displaced liquid

or, VDg = vdg or, \(\frac{v}{V}\) = \(\frac{D}{d}\)

ii) If n is the fraction of volume of the body that remains immersed in the liquid, then

\(\frac{v}{V}\) = n = \(\frac{D}{d}\) or, D = n × d

### Application and Illustration of the Principle of Floatation

The upward motion of a balloon: The upward motion of a balloon depends on the upthrust of air. The net weight of an inflated balloon containing hydrogen gas is much less than the weight of air displaced by it (due to lower density of hydrogen). As a result, the buoyant force exerted by air on the balloon becomes greater than the weight of the balloon. Due to this reason, the balloon experiences a net upward force and it moves up. We know that with the increase in altitude from the surface of the earth, the density of air decreases and hence the upthrust exerted on the balloon decreases.

After attaining a certain height, the weight of the balloon becomes equal to the buoyant force acting on it and then the balloon cannot rise any further; it remains floating at a particular altitude. Again, another situation may arise. Due to the lower atmospheric pressure at a higher altitude, the volume of the balloon gradually increases. As a result, the weight of air displaced by the balloon, i.e., the upthrust also increases and, in consequence, the balloon rises more and more and finally may burst due to excessive expansion in its volume.

Besides hydrogen, helium is also used as a filling agent in a balloon. Hydrogen is lighter than helium, but is inflammable whereas helium is not. For this reason, helium is used in balloons.

Why does Ice float on water?

The mass of 1 cm^{3} of ice is 0.917 g. The mass of 1 cm^{3} of water is 1 g. So, the same volume of ice is lighter than water. Since the density of ice is less than that of water, it floats partially immersed in water. If nth part of a lump of ice remains submerged, then n = density of ice = 0.917 = \(\frac{11}{12}\) (approx.)

So, in its floating conditions \(\frac{11}{12}\)th part of a lump of ice remains immersed in water and \(\frac{1}{12}\)th part remains above the water surface. The density of sea water is 1.03 g ᐧ cm^{-3}. So, \(\frac{1}{9}\)th part (approx.) of an iceberg remains exposed to air while floating on sea water.

To float on water, we have to learn swimming but to animals swimming comes naturally-explain why.

The average density of the human body is 1.01 g ᐧ cm^{-3} (approx). The weight of the body of a man is less than the weight of an equal volume of water. But the head is heavier than the weight of an equal volume of water. Hence, a man can float on water, but his head sinks in it. So, we have to learn swimming to float on water. A man strives to keep his head afloat by moving his limbs in water. This is the act of swimming. For animals, both their heads and bodies are lighter than an equal volume of water and hence they can float on water effortlessly.

It is easier to float in sea water than in fresh river water because the density of sea water is more than that of river water. Hence, the upthrust of sea water is more than that of river water.

A piece of iron sinks into water but a ship made of iron floats on it—explain how.

This is due to the special shape of a ship. The bottom of a ship is made wider and hollow. It is given a concave shape so that the weight of displaced water by the ship becomes equal to the weight of the ship. According to the condition of floatation, if the weight of a body becomes equal to the upthrust exerted on it by the displaced liquid then the body floats on that liquid. That is why a ship floats on water. The weight of a piece of iron is more than the weight of water displaced by it and so it sinks in water.

How does a submarine submerg and float on water?

A submarine can float on water, but can also be submerged. A submarine contains a large number of tanks which can be filled up with air or water. These tanks are called ballast tanks. When the ballast tanks are filled with air, the submarine floats on water as the weight of the submarine becomes less than the weight of water displaced by it. When water is admitted into the ballast tanks using a pump, the submarine becomes slightly heavier than the water

displaced by it and hence it submerges. To raise the submarine again to the surface of water, water is driven out of the ballast tanks by compressed air.

Why are life-belts used?

Steamers and ships are provided with life-belts, If a steamer or a ship sinks, then the passengers can float by wearing these life-belts. A life-belt is nothing but an air-filled bag. The weight of this bag along with the weight of the person is less than the weight of the displaced water and so the person is able to keep afloat.

1 kilogam of cotton is heavier than 1 kilogram of iron: For a body in air,

apparent weight = actual weight (i.e., weight in vacuum) – buoyancy of air

So, actual weight apparent weight + buoyancy of air = apparent weight + weight of air displaced by the body

When we weigh 1 kg of cotton and 1kg of iron in air, those are their apparent weights. So the actual weight depends on the weight of air displaced by them.

Clearly 1kg of Cotton has a much larger volume, and it displaces a much larger amount of air, S0 the actual weight

of 1kg of cotton is higher than that of 1kg of iron. Therefore, it is said that, 1kg of cotton is heavier than 1kg of iron.