JEE Main & Advanced Physics Fluid Mechanics, Surface Tension & Viscosity / द्रव यांत्रिकी, भूतल तनाव और चिपचिपापन Archimedes Principle

Accidentally Archimedes discovered that when a body is immersed partly or wholly in a fluid, at rest, it is buoyed up with a force equal to the weight of the fluid displaced by the body. This principle is called Archimedes principle and is a necessary consequence of the laws of fluid statics.

When a body is partly or wholly dipped in a fluid, the fluid exerts force on the body due to hydrostatic pressure. At any small portion of the surface of the body, the force exerted by the fluid is perpendicular to the surface and is equal to the pressure at that point multiplied by the area. The resultant of all these constant forces is called upthrust or buoyancy.

To determine the magnitude and direction of this force consider a body immersed in a fluid of density \[\sigma \] as shown in figure. The forces on the vertical sides of the body will cancel each other. The top surface of the body will experience a downward force.

\[{{F}_{1}}=A{{P}_{1}}=A({{h}_{1}}\sigma g+{{P}_{0}})\]                      [As \[P=h\sigma g+{{P}_{0}}\]]

While the lower face of the body will experience an upward force.

\[{{F}_{2}}=A{{P}_{2}}=A({{h}_{2}}\sigma g+{{P}_{0}})\]

As \[{{h}_{2}}>{{h}_{1}},\,{{F}_{2}}\] will be greater than \[{{F}_{1}}\], so the body will experience a net upward force

\[F={{F}_{2}}-{{F}_{1}}=A\sigma g({{h}_{2}}-{{h}_{1}})\]

If L is the vertical height of the body \[F=A\sigma gL=V\sigma g\]                    

[As \[V=AL=A({{h}_{2}}-{{h}_{1}})]\]

i.e., F = Weight of fluid displaced by the body.

This force is called upthrust or buoyancy and acts vertically upwards (opposite to the weight of the body) through the centre of gravity of displaced fluid (called centre of buoyancy). Though we have derived this result for a body fully submerged in a fluid, it can be shown to hold good for partly submerged bodies or a body in more than one fluid also.

(1) Upthrust is independent of all factors of the body such as its mass, size, density etc. except the volume of the body inside the fluid.

(2) Upthrust depends upon the nature of displaced fluid. This is why upthrust on a fully submerged body is more in sea water than in fresh water because its density is more than fresh water.

(3) Apparent weight of the body of density \[(\rho )\] when immersed in a liquid of density \[(\sigma )\].

Apparent weight = Actual weight - Upthrust \[=W-{{F}_{up}}\]

\[=V\rho g-V\sigma g=V(\rho -\sigma )g\]\[=V\rho g\left( 1-\frac{\sigma }{\rho } \right)\]

\[\therefore \]   \[{{W}_{APP}}=W\left( 1-\frac{\sigma }{\rho } \right)\]

(4) If a body of volume V is immersed in a liquid of density \[\sigma \] then its weight reduces.

\[{{W}_{1}}\] = Weight of the body in air,   \[{{W}_{2}}\] = Weight of the body in water

Then apparent (loss of weight) weight \[{{W}_{1}}-{{W}_{2}}=V\sigma g\] 

\[\therefore \] \[V=\frac{{{W}_{1}}-{{W}_{2}}}{\sigma g}\]

(5) Relative density of a body

(R.D.) =\[\frac{\text{density of body}}{\text{density of water}}\] \[=\frac{\text{Weight of body}}{\text{Weight of equal volume of water}}\]= \[\frac{\text{Weight of body}}{\text{Water thrust }}\]

\[=\frac{\text{Weight of body}}{\text{Loss of weight in water}}\]

= \[\frac{\text{Weight of body in air}}{\text{Weight in air--weight in water}}\] = \[\frac{{{W}_{1}}}{{{W}_{1}}-{{W}_{2}}}\]

(6) If the loss of weight of a body in water is \['a'\] while in liquid is \['b'\]

\[\therefore \] \[\frac{{{\sigma }_{L}}}{{{\sigma }_{W}}}=\frac{\text{Upthrust on body in liquid}}{\text{Upthrust on body in water}}\]

\[=\frac{\text{Loss of weight in liquid}}{\text{Loss of weight in water}}\]\[=\frac{a}{b}=\frac{{{W}_{\text{air}}}-{{W}_{\text{liquid}}}}{{{W}_{\text{air}}}-{{W}_{\text{water}}}}\]