Upthrust and Law of Floatation for JEE

When we push an object inside the water, we always experience there is some resistance in pushing it down. Let’s take one more scenario. When we swim in the water, we feel lighter, i.e., there’s an apparent loss in weight. Have you ever wondered how that happens? 

The reason is that when an object is pushed inside any fluid or simply submerged, then the fluid exerts an upward force on the object that is termed as ‘Upthrust’, also known as ‘Buoyant Force’. Some of the characteristics of Buoyant force are listed below.

• When any object is immersed in a fluid (partially or wholly), an upward force is exerted on it.

• This upward force exerted on the lower surface of the object is known as upthrust or buoyant force.

• The upthrust is denoted by F_B and the unit of measurement is Newton (N) or kgf.

Definition of Buoyancy

The property of a fluid to exert an upthrust or buoyant force on an object immersed in it is known as buoyancy. Both liquids and gases possess this property. 

Some of the examples are: we experience an upward resistance while pushing any object inside the water and a hydrogen filled balloon rises up in air due to the upthrust experienced.

• The condition that an object will float in a liquid is that the weight of the object (W) should be less than the buoyant force exerted by the liquid.

• The condition that an object will sink in a liquid is that the weight of the object (W) should be greater than the buoyant force exerted by the liquid.

Archimedes’ Principle

According to Archimedes’ principle, ‘If a body is completely or partially immersed in a fluid, then it experiences an upwards force or an upthrust, which results in an apparent loss of weight equal to the weight of the fluid displaced by the body.’ It is the first condition of equilibrium. It is assumed that the upthrust mentioned above, also known as the buoyant force, is located at the submerged part’s centre, which is named the centre of buoyancy. The story behind Archimedes' principle is that Archimedes discovered this law while he was taking a bath. 

Any object immersed in a fluid is subjected to an upthrust because the pressure exerted on the bottom surface of the immersed object by the fluid is greater than the pressure exerted on the upper surface since the pressure increases with an increase in depth.

$p=h\rho g$, 

where ‘h’ is the height of the fluid column,

‘ρ’ (rho) is the density of the fluid, and

‘g’ is the acceleration due to gravity

Determination of Relative Density by Archimedes Principle

Definition of Relative Density

Relative density or specific gravity of a substance is defined as the ratio of the density of the substance to the density of water. It is also equal to the weight of a substance to the weight of water of an equal volume. Since weight is directly proportional to mass, therefore, it can also be defined as the ratio of the mass of a substance to the mass of water of an equal volume.

• Relative density of a solid: The weight of the body in air is w₁, and when it is completely immersed in water, its weight becomes w₂. Then, the relative density of the solid will be given as:

Relative density of solid $=\frac{Weight \ \ of \ \ solid \ \ in \ \ air }{ Weight \ \ of \ \ equal \ \ volume \ \ in \ \ water}$  

i.e., Relative density of solid $=\frac{{{w}_{1}}}{{{w}_{1}}+{{w}_{2}}}$

• Relative density of a liquid: The weight of the body in air is w₁, and when it is completely immersed in water, its weight becomes w₂. If the same body is immersed in a given liquid, then, its weight becomes w₃. Then, the relative density of the liquid will be given as: 

Relative density of liquid $=\frac{apparent \ \ loss \ \ of \ \ weight \ \ of \ \ solid \ \ in \ \ liquid}{apparent \ \ loss \ \ of \ \ weight \ \ of \ \ solid \ \ in \ \ water}$

i.e., Relative density of liquid $=\frac{{{w}_{1}}-{{w}_{3}}}{{{w}_{1}}-{{w}_{2}}}$

Law of Floatation

Floatation

Floatation is defined as the property of a body to rise or stay up in the upper levels of a fluid when it is immersed in it. It is just the opposite of sinking which is defined as the property of a body to move to the lower levels of the fluid when immersed in it.

The Law of Floatation states that a floating body displaces its own weight of the fluid in which it floats. 

Conditions for a body to float: The conditions that a body will float in a fluid are listed below.

• The density of the fluid in which the body is immersed should be greater than that of the body itself.

• The upthrust force exerted by the fluid on the lower surface of the body must be equal to the weight of the body.

• The volume of the body should be large enough to displace a large amount of fluid.

Relation between Upthrust and the Real Weight of the Body

• The real weight of the body will be equal to the apparent weight of the body when it is immersed in fluid added to the apparent loss in weight.

• The apparent loss in weight is equal to the weight of the fluid displaced or upthrust, according to the Archimedes Principle. 

• Therefore, the real weight would be equal to the Apparent weight of the body inside a fluid added to the upthrust. 

• We know that the apparent weight of a body becomes zero when it floats in a fluid. Therefore, while floating the real weight of the body becomes equal to the Upthrust.

Therefore, the relationship between upthrust and the real weight of a body is that while floating, the real weight of the body is equal to the upthrust exerted by the fluid. The above result leads to the Law of Floatation, that is, a floating body displaces its own weight of the fluid in which it floats.

Summary

• When any object is immersed in a fluid (partially or wholly), an upward force is exerted on it, known as Upthrust or Buoyant force.

• The upthrust is denoted by F_B and the unit of measurement is Newton (N) or kgf.

• According to Archimedes’ principle, ‘If a body is completely or partially immersed in a fluid, then it experiences an upward force or an upthrust, which results in an apparent loss of weight equal to the weight of the fluid displaced by the body.

• Relative density or specific gravity of a substance is defined as the ratio of the density of the substance to the density of water.

• The Law of Floatation states that a floating body displaces its own weight of the fluid in which it floats.