Class 11 > Unit # 06: Fluid Statics > Archimedes Principle & Upthrust

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Fluid Mechanics - Archimedes' Principle & Upthrust

Q. State and explain Archimedes’ Principle. Derive expression for upthrust.

Archimedes' Principle

"Archimedes' Principle states that when an object is immersed partially or wholly into a fluid, it experiences an upward buoyant force (upthrust) which is equal in magnitude to the weight of the fluid displaced by that object."

Explanation

The magnitude of the upward buoyant force depends upon two key factors:

1. Volume of the Body: The more fluid that is displaced by the object, the greater the upthrust it experiences.
2. Density of the Fluid: The greater the density of the fluid, the greater the resulting upthrust.

A floating body displaces its own weight of fluid such that there is zero vertical resultant force acting on the body. Therefore, if a sphere of material density $\rho$ and radius $r$ is fully immersed into a liquid of density $\rho_f$, its apparent weight is reduced by the buoyant force.

Figure: Cylindrical object immersed in a liquid vessel experiencing fluid pressure difference.

Upthrust (Buoyant Force)

Upthrust is the net upward force exerted on an object submerged in a fluid, equal in magnitude to the weight of the displaced fluid. This force supports the object and prevents it from sinking.

As an object dips deeper into a fluid, hydrostatic pressure increases with depth ($P = \rho_f g h$). Consequently, the bottom surface of a submerged object experiences greater upward pressure than the top surface experiences downward pressure. This pressure differential produces a net upward force called upthrust (also known as buoyancy or buoyant force).

Mathematical Derivation of Upthrust

Consider a cylinder of height $h$ and cross-sectional area $A$ immersed vertically in a vessel containing a liquid of density $\rho_f$. The net upthrust ($F_u$) acting on the cylinder from bottom to top is given by:

$F_u = - (F_2 - F_1)$    --- (i)

We know that fluid pressure is given by $P = \rho_f g h$, and force is $F = P \times A$. Substituting pressure into the force terms:

$F_u = - (P_2 A - P_1 A) = - (\rho_f g y_2 A - \rho_f g y_1 A)$

Factoring out the common terms:

$F_u = - \rho_f g A (y_2 - y_1)$

Note on signs: The negative sign indicates that upthrust acts in the upward direction, whereas standard fluid gravitational pressure acts downward.

Since the height of the cylinder is $h = y_2 - y_1$, and the volume of the cylinder (and thus the displaced fluid) is $V = A \times h$:

$F_u = \rho_f g (A \cdot h)$

$F_u = \rho_f g V$

Where:

• $F_u$ = Upthrust or Buoyant Force
• $\rho_f$ = Density of the fluid
• $g$ = Acceleration due to gravity (acting vertically downwards, producing upward reaction)
• $V$ = Volume of the displaced fluid / submerged body

Since $\rho_f \times V$ represents the mass of the displaced fluid ($m_f$), $\rho_f V g$ equals the weight of the displaced fluid, perfectly proving Archimedes' Principle.

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