Check our notes on Pascal’s law of pressure and hydraulic machines from here. In this part, we will study about upthrust, Archimedes’ principle, and floatation.
Upthrust
Activity: Observation of Upthrust
Take an empty bottle and some water in a bucket. Tighten the cap of the bottle to make it alright. Drop the bottle into the bucket. The bottle floats on the water. Push the bottle into the water. Observe, discuss, and answer the following questions.
a. When the bottle is pushed into the water, do you feel that the bottle is being pushed upwards?
b. When the plastic bottle is let go in the air it falls because of gravity, but why doesn’t it sink when it is placed on the surface of the water? Doesn’t gravity act on the bottle when it is placed in water?
It is difficult to push the bottle into the water. It means the water is pushing the bottle upward. As the bottle is pushed into the water, the upward force exerted by the water goes on increasing until the bottle is completely submerged. When the bottle is released, it returns to the surface of the water. The net upward force by which water pushes the object in it is called upthrust. All fluids exert upthrust but the magnitude of the upthrust depends upon their density.
When an object is placed in air or water, it is acted upon by two forces; weight caused by gravity and the upthrust of the liquid. Since both these forces are in the opposite direction, the upthrust reduces or neutralizes the weight.
Therefore, when an object is partially or completely immersed in a gas or liquid (fluid), the upward force acting on the object is called upthrust. Its SI unit is Newton (N).
The upthrust of the air is negligible as compared to the upthrust of the water. In all cases, the upthrust caused by the air is not negligible. Aeroplanes experience a large upthrust of the air, which helps to balance the weight.
Cause of upthrust
When an object is immersed in water, it experiences normal pressure in all directions. The forces in the opposite directions are canceled on all the other sides rather than the top and bottom of the object, as the normal pressure exerted by the liquid is equal.
But, as the pressure increases as the depth of the object increases, the lower (bottom) surface of the object is greater than the top surface, so the bottom surface experiences more force than the top surface, which cannot be completely balanced by the force on the top surface. This, when the force on the upper surface of the object is subtracted from the force on the bottom surface, the resultant force has an upward direction. Thus, the resultant force is known as upthrust and its direction is always upward.
Activity: Calculation of the upthrust acting on a stone immersed in water
Take a small stone, a beaker that is filled more than half with water, a spring balance, and a piece of thread. Tie the stone with the thread and hang it on the spring balance. Measure the weight of the stone when it is suspended in air and the weight of the stone when it is submerged in water. Is the weight in air equal to the weight in water?
When an object is immersed in a liquid, its weight is found to be decreased due to the upthrust exerted on it. The weight of an object measured in air (W1) is called real weight and the weight of an object inside a liquid (W2) is called apparent weight.
Therefore, the upthrust created when an object is immersed in a liquid can be represented by the following formula:
Upthrust (U) = Actual Weight (W1) – Apparent Weight (W2)
Factors Affecting Upthrust
There are two major factors affecting the upthrust. They are described below.
a. Density of a liquid
Activity: Observation of the weight of stones in different media
Prediction: In which liquid, is the weight of the stone least, when it is immersed in water, glycerin, and cooking oil respectively?
Take some amount of water and edible oil in two beakers. Hang the stone on a spring balance and measure the weight of the stone in these three liquids.
| Weight in water | Weight in edible oil | Weight in glycerine | Results |
| – – | – – | – – | – – |
The density of the edible oil and water is 0.90g/cm3 and 1g/cm3 respectively. In the above activity, the weight of the stone is less in water than that in edible oil. Since the density of water is more than that of edible oil, the upthrust exerted by the water is also more than that of edible oil. Thus the upthrust exerted on a body is directly proportional to the density of the liquid. Therefore, when an object is immersed first in a liquid with lower density and then in a liquid with higher density, the upthrust exerted by the liquid of the higher density is more.
Upthrust ∝ Density of liquid
Since the upthrust is directly proportional to the density of the liquid, the iron ball sinks in water but floats on the surface of mercury which has a high density. Similarly, when an egg is placed in a glass filled more than half with tap water, it sinks. But when salt is dissolved in the water, the egg starts to float. The density of salty water is higher than that of tap water.
That’s why salty water exerts more upthrust than tap water. As a result, the egg sinks in tap water but floats on salt water. This is the reason why it is easy to swim in seawater. Similarly, the ship sinks less in seawater than in river water.
b. Volume of liquid displaced
When a solid object is immersed in a liquid, as its volume inside the liquid increases, the upthrust on it also increases and when the object is completely immersed in the liquid, the upthrust is maximum. After the object is immersed in the liquid, even though the depth of the object increases, the upthrust remains constant. For example, two cubic solids of different weights but equal volume are completely immersed in a liquid; the upthrust acting on them is the same. Therefore, when an object is partially or completely immersed in a liquid, the upthrust acting in it is directly proportional to the volume of the liquid displaced. This means, when the volume of an object immersed in liquid increases, the upthrust acting on it also increases.
Upthrust ∝ Volume of fluid displaced
When trying to submerge a big water-tight plastic bottle and a small water-tight plastic bottle, the big bottle displaces more water than the smaller one. As a result, a large plastic bottle experiences more upthrust. Therefore, to submerge a large plastic bottle in water, more force should be applied than on the small bottle. A similar effect applies to the large and small balloons filled with helium gas. Since the large balloon displaces more air then the small balloon, the upthrust acting on it is also greater.
When an object is placed in a fluid, the relationship between the upthrust on the object and the weight of the displaced fluid is given by Archimedes’ principle.
Archimedes’ Principle
Activity: Verification of Archimedes’ Principle
Take a spring balance, a eureka can (beaker that can collect displaced water), a beaker, a small stone, and a mug. Hang the stone on a spring balance and measure its weight first in the air and then water in the eureka can. While measuring the weight of the stone in the water, note the weight of the water displaced by the stone too. For this, weigh the empty beaker and subtract the weight of the beaker with displaced water.
The data required for this activity can be collected in the table as below.
| weight of the stone in the air (w1) | weight of stone in water (W2) | Upthrust (U)= w1-w2 | Weight of the empty beaker (w3) | Weight of beaker and displaced water (w1) | Weight of the displaced water (w1) | Result |
In this process, when a stone is immersed in water, the upthrust exerted on it is equal to the weight of the displaced water. This proves Archimedes’ principls.
Archimedes is a Greek mathematician born in 287 BC. According to Archimedes’ principle, when an object is partially or completely immersed in a liquid, the upthrust is equal to the weight of the liquid displaced by it. This principle is not only true for liquids but applies to all fluids.
Mathematically,
Upthrust (U) = Weight of liquid displaced (W)
U = mg =V ρ g ( density of the liquid (ρ = Mass (m)/Volume (v).)
In the above equation V’, ρ’, and g represent the volume of liquid displaced, density, and acceleration due to gravity respectively.
The upthrust exerted on an object placed in a liquid depends on the volume of liquid displaced by the object. Likewise, according to Archimedes’ principle, the upthrust on an object is equal to the weight of the displaced liquid. Therefore, when an object is placed in a liquid, whether it floats or sinks depends on its weight of the object and the upthrust which depends on the volume of the liquid displaced.
Floatation
If the density of the object is less than the density of the liquid, the object will float on the liquid. But, simply comparing the density of an object to the density of a liquid does not determine whether an object floats or sinks.
Activity: Observation of the state of sinking and floating
Take two pieces of aluminum foil of equal area and a beaker that is filled more than half with water. Squeeze one of the aluminum foils in the shape of a ball. Then drop both aluminum pieces into the water. Observe which floats and which sinks. If the aluminum foil is turned into a ball, its surface gets reduced. Since it displaces less water, the upthrust acting on it will be less than its weight. So, it sinks in water. If the aluminum foil is placed in the water as a sheet, the surface area increases and so does the upthrust acting on it. When weight and upthrust become equal, the foil floats.
The density of aluminum is 2700 Kg/m3 which is 2.7 times the density of water. Thus, it can be concluded that the foil sinks in water. But, as the sheet floats on the water too. Therefore, whether an object floats or sinks in a liquid depends not only on the density od the substance but also on the shape of the object, i.e. the resultant force acting on an object when it is placed in the liquid.
The weight of an object and the upthrust are the forces acting on the object in exactly opposite directions. When the upthrust cancels the weight of the object, the resultant force becomes zero and hence the object floats in the liquid.
An object floating in a liquid may be partial if completely inside the surface of the liquid. When partially floating, only the submerged part of the object displaces the liquid. For an object to float in a liquid, the weight of the object must be equal to the weight of the liquid that it displaces. This is known as the law of floatation. This means, for an object to float on a liquid,
Weight of object = Weight of liquid displaced
This fact applies to various phenomena that occur in the liquid and air around us. Some examples are given below.
a. Floating in liquid
When an iron block is placed in water, it cannot displace the volume of the water whose weight is equal to its weight, and thus, it sinks. If the iron block is shaped like a bowl, the bowl can displace water equal to its weight. Similarly, in a ship, its hull is to be made wide, long, and deep. Because of this, the ship can displace enough water to generate upthrust equal to its weight. For example, to prevent a ship weighing 50000 tons from sinking, the size of the hull should be able to displace 50000 tons of water. Therefore, iron sinks in water, but a ship made up of iron floats in water.
Submarines are built in such a way that they can float visibly on the surface of the water as well as inside the water. When its blast tank is filled with water, the weight increases and the submarine goes deeper into the water, and when the water is blown out of the tank, the weight decreases and it floats back on the surface.
Like submarines, fish also use Archimedes’ principle to float and sink in water. When air is filled in the fish’s swim bladder i.e. air sac, the volume of the body increases and upthrust also increases, the fish floats towards the surface of the water. On the contrary, fish empty their bladders to reduce their body volume as well as the upthrust acting on their body. This helps them get deeper into the water.
A hydrometer is a device used to measure the relative density of liquids. It contains a wide glass bulb filled with heavy metal which is attached to a calibrated fine tube. Due to the liquid displaced by its heavy bulb, it gets the necessary upthrust and floats vertically on the surface of the liquid. The hydrometer floats more on the liquid of high density because such liquid exerts more upthrust. On the contrary, if the density of the liquid is low, the hydrometer sinks more. Lactometer is a type of hydrometer used to test the mixture of water in milk.
b. Floating in the atmosphere
Like in the water, there is upthrust in the air too. But the density of air is much less than that exerted by water on an object.
Why does a balloon filled with helium float in the air?
Objects can float on gases as well as on liquids. Objects float or fly in the air due to the upthrust produced by the air pressure.
In the case of a balloon filled with helium gas, the surrounding air is denser than the helium. If the weight of the air is displaced by the helium-filled balloon, the balloon will fly upwards. As the altitude increases, the density of the air decreases and this makes the upthrust acting on the balloon decrease. When it reaches a certain height, the weight of the balloon equals the upthrust. As a result, the balloon floats in the air. At a certain, the pressure inside and outside the balloon becomes unbalanced and the volume of the balloon keeps increasing until it finally bursts.
Like hydrogen or helium balloons, hot air balloons also fly in the air. The density of air inside a hot air balloon is less than the density of the surrounding air. Since such a balloon displaces a large amount of air, the upthrust on the balloon becomes greater than its weight. As a result, the balloon flies upwards in the air. When the balloon reaches a maximum altitude, the weight of the displaced air i.e., upthrust balances the weight of the balloon. Then the balloon floats in the air.
A balloon filled with hot air rises or falls in the air and such a condition is controlled by changing the temperature of the air inside the balloon with the help of a burner. When the flame from the burner heats the air in the balloon, the density of the air decreases. This causes the balloon to fly upwards. On the contrary, when the burner is turned off, the air in the balloon cools, and its density increases. When the air in the balloon is cold enough, the weight of the balloon becomes greater than the upthrust on it, and the balloon sinks in the air. i.e. descends to the ground.
Some Important Questions
Q. A wooden cork dipped below the surface of the water by applying force with a finger comes to the surface when the finger is removed. Give Reason
Ans. It is because of the upthrust provided by the water. The upthrust provided by the water is in the upward direction and when we apply the force with a finger, the resultant force is downward, so the cork dips, but as soon as we remove the finger the resultant upthrust force acts in the upward direction and hence the cork comes to the surface of the water.
Q. On lifting a stone submerged in water, it feels heavier when it comes out of the water. Give Reason.
Ans. When lifting a stone under water, the upthrust acts in an upward direction, which makes the stone lighter, but as soon as the stone comes out of the water, it feels heavier because the upthrust provided by the air is negligible than the upthrust provided by the water.
Q. Write any two applications of Archimedes’ principle.
Ans. The two applications of Archimedes’ principle are :
- Floating of a submarine on the surface and inside the water.
- Flying off a hot-air balloon in the air.
Q. Differentiate between Pressure and Upthrust.
Ans.
| Pressure | Upthrust |
| Upward force acting perpendicular on a given surface. | The upward force exerted by fluid acting perpendicular on a given surface, when an object is immersed fully or partially in that fluid. |
| P=F/A | U= vol of fluid displaced by an object |
Q. If more passengers climb a boat than its maximum capacity, the boat is likely to sink. Give Reason
Ans. When more passengers climb a bot than its maximum capacity, the weight of the boat increases, and the boat starts to immerse, as the boat immerses up to the surface, upthrust will be maximum and no further volume is displaced, which makes the boat likely to sink.
Q. State the law of floatation.
Ans. The law of floatation states that ” For an object to float in a liquid, the weight of the object must be equal to the weight of the liquid that it displaces.”
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