Deriving the Dimensional Formula for Surface Tension: A Step-by-Step Guide for SEO

Understanding the dimensional formula for surface tension is crucial for various scientific and engineering applications. In this article, we will walk through a detailed step-by-step method to derive the dimensional formula using the given formula ( s frac{pgrh}{2} ), focusing on the foundational concepts that are easily digestible for SEO purposes.

Introduction to Surface Tension and the Given Formula

Surface tension is a property exhibited by the surface of a liquid that allows it to resist external forces. It is a direct consequence of the cohesive forces between liquid molecules. One common formula used to understand surface tension involves pressure ( p ), acceleration due to gravity ( g ), radius ( r ), and height ( h ).

The formula given is:

( s frac{pgrh}{2} )

Let's break it down into understandable components and derive the dimensional formula step-by-step.

Identifying the Dimensions of Each Variable

Pressure ( p )

Pressure is defined as force per unit area.

( [p] frac{F}{A} )

Force is mass times acceleration, and area is length squared:

( [p] frac{MLT^{-2}}{L^2} ML^{-1}T^{-2} )

Acceleration due to Gravity ( g )

Acceleration due to gravity is given by length per time squared:

( [g] frac{L}{T^2} )

Radius ( r )

The radius is simply a length:

( [r] L )

Height ( h )

Height is also a length:

( [h] L )

Substituting Dimensions into the Formula

Now, we substitute the dimensions of each variable into the formula for surface tension ( s ):

( s frac{pgrh}{2} )

Ignoring the constant ( frac{1}{2} ) for dimensional analysis, we have:

( [s] [p] cdot [g] cdot [r] cdot [h] )

Substituting the dimensions:

( [s] left( frac{M}{L^3} right) cdot left( frac{L}{T^2} right) cdot L cdot L )

This simplifies to:

( [s] frac{M}{L^3} cdot frac{L}{T^2} cdot L cdot L frac{M cdot L^3}{L^3 cdot T^2} frac{M}{T^2} )

Conclusion

The dimensional formula for surface tension ( s ) is:

( [s] frac{M}{T^2} )

This method of dimensional analysis is a powerful tool not only for surface tension but for understanding the physical properties of various phenomena. By breaking down the units and applying basic algebra, we can derive the dimensional formulas for any given physical relationship.

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