Archimedes Principle: Formula & Apparent Weight

Archimedes Principle explains the relationship between the apparent weight of the object immersed in water and the weight of the water that is displaced by it. It includes factors like Buoyant Force which helps establish the relationship. Archimedes’ Principle, or the physical law of buoyancy, was discovered by Greek mathematician Archimedes. The formula of Archimedes’ Principle is: F_b = ρ x g x V.

Here,

• F_b = buoyant force
• ρ = density of the fluid
• V = submerged volume
• g = acceleration due to gravity

Archimedes’ principle states that a body at rest, completely or partially submerged in fluid, is acted upon by a buoyant force, the magnitude of which is equivalent to the weight of displaced fluid. This is considered the first condition of equilibrium. The displaced portion of the fluid's weight is equal to the magnitude of the buoyant force. One example of where its application is used are Hydrometers, which is based on the principle of Archimedes.

Key Terms: Archimedes’ Principle, Buoyant force, Apparent Weight, Density, Hydrometers, Equilibrium, Force, Water

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What is Archimedes’ Principle?

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Archimedes Principle claims that:

Archimedes Principle

The value of the thrust force in the upward direction can be obtained by Archimedes' principle. The apparent loss of weight will be equivalent to the weight of the liquid displaced when an object is either fully or partially immersed in a liquid.

Note: Archimedes Law is known to act in the upward direction at the centre of mass of the fluid that is displaced.

What is Buoyant Force?

Buoyant force can be defined as:

• Buoyant force can be defined as an upward force responsible for the apparent decrease in immersed object's weight.
• The decrease in the object’s weight is equivalent to the water it displaced.
• Buoyancy can be found in any object which has a density greater than that of the fluid wherein it is immersed.

What is Archimedes' principle used for?

Archimedes’ principle is mainly used for:

• Archimedes' principle is used to calculate the volume of an object which doesn’t have a regular shape. That means, the irregular-shaped object can be submerged, and the volume of the fluid displaced will be equivalent to the volume of the object.
• It is also widely used to determine the density or specific gravity of an object.

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Archimedes’ Principle Formula

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The thrust force, also known as the buoyant force, is responsible for enabling objects to float. Thus, this equation can also be called the “law of buoyancy”. Mathematically, the formula for Archimedes Principle can be given as:

Here,

• F_b = buoyant force
• g = acceleration due to gravity
• p = density 
• V = volume

Archimedes’ Principle Experiment

Archimedes’ principle can be best shown by an experiment as listed below:

• Take any container which has water filled to the brim.
• Take a solid object and measure its weight by using a spring balance. 
• Now, keep the object attached to the spring balance and start immersing it in the water. Do not immerse the spring balance.
• Note the weight down as is shown by the spring balance. You can now notice that it is lesser. Meaning that some water is displaced into the bowl.
• Collect the displaced water and weigh it.
• Thus, you will notice that the weight of the water is equivalent to the loss of weight of the object.

Archimedes’ principle Experiment

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Archimedes’ Principle Derivation

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We know that Density can be expressed as: \(\begin{array}{l}Density\,(\rho )=\frac{Mass\,(M)}{Volume\,(V)}\end{array}\)

Hence, the displaced liquid’s mass can be shown as:

Mass (M) = Density (\(\rho\)) \(\times\)Volume (V)

So, weight of the displaced liquid can be evaluated as:

Weight = Mass \(\times\) Acceleration due to Gravity

\(\therefore\) Weight = Mass \(\times\) g = \(\rho\) \(\times\)V \(\times\) g

From Archimedes’ Principle, we are aware that the apparent loss of weight is equivalent to the weight of displaced water. It can be written as:

Mathematically, thrust force can be now expressed by the following equation:

Thrust Force = \(\rho\) \(\times\) V \(\times\) g

or, F_b = p × V × g

where, 

• F_b = buoyant force, 
• p = density of the fluid, 
• V= volume of fluid
• g = acceleration due to gravity.

The video below explains this:

Archimedes Principle Detailed Video Explanation:

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What is Apparent Weight?

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Apparent weight can be defined as a property of objects which corresponds to how heavy the object originally is.

• The apparent weight of an object differs from the original weight of the object when the force of gravity which acts on the object is not balanced by an equal but opposite force.
• The actual weight of an object, if placed on a solid surface, begins to act in a downward direction via the centre of gravity.
• When this object is immersed in water, it experiences an upward force which is the buoyant force. 
• This buoyant force decreases the downward force of the object to some extent and the object feels lighter, which is the apparent weight of the object.

In order to calculate the apparent weight, one can subtract buoyant force from the actual weight of the object. Thus,

Apparent Weight = Buoyant Force – Actual Weight

i.e., Apparent Weight Formula, a = dv/dt

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Laws of Floating

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When an object is immersed in water, it either floats or sinks based on the density of the object and the liquid. Let us consider W¹ as the actual weight of the object and W² as the buoyant force.

• When W¹>W², the density of the object is more than that of the liquid causing it to sink.
• When W¹=W², the density of the object and the liquid are equal and the object will float in the water at any depth and remain completely submerged in the liquid.
• When W¹< W², the density of the object is less than that of the liquid causing it to float(partially submerged). 

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Archimedes’ Principle Applications

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Archimedes' Principle has several real-life applications. 

• Archimedes' Principle helps to calculate the volume and density of objects. The proportions of constituent metals in an alloy can be determined with the help of the Archimedes Principle.
• Ships float on water due to Archimedes' Theory. The ships are made hollow so that their density is less than the water and therefore they remain only partially submerged in water.
• The depth at which submarines float is determined through Archimedes' Principle. Submarines have a ballast tank in which water is filled. The weight of the submarine can be increased or decreased by varying the amount of water in the ballast tank which is then used to fix the depth at which it floats.
• Hydrometers, that are used to calculate the density of the liquid work on Archimedes’ Principle.

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Things to Remember

• Archimedes’ Principle discovered by Archimedes, explains the relationship between the apparent weight of an object submerged in water and the weight of the water displaced by it.
• Buoyant force is an upward force responsible for the apparent decrease in the weight of the immersed object.
• The Formula for Archimedes’ Principle can be given as F_b = ρ x g x V (here, F_b = buoyant force, g = acceleration due to gravity, \(\rho\) = density, V = volume.)
• When an object is immersed in water, it either floats or sinks based on the density of the object and the liquid.
• Archimedes’ Principle has several real-life applications. Ships and submarines float/submerge based on this principle. Volume and Density of objects can also be calculated through it.

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