Surface Tension

The given formula is:

Where:

• h is the height of the fluid column,
• T is the surface tension,
• r is the radius of the tube,
• ρ is the density of the fluid,
• g is the acceleration due to gravity,
• θ_c is the contact angle between the fluid and the solid surface.

The equation shows that the height h is directly proportional to the cosine of the contact angle θ_c and the density of the fluid ρ.

The relationship can be expressed as:

Which means, the cosine of the contact angle is directly proportional to the density of the fluid.

Comparing Densities and Contact Angles:

If ρ₁ > ρ₂ > ρ₃, then the corresponding contact angles will follow the order:

Thus, the angles will satisfy the relation:

In other words, as the density increases, the contact angle decreases.

Step 1: Use the Formula for Force Due to Surface Tension

The force required to overcome surface tension is given by:

$ = Force due to surface tension

= Radius of the liquid surface

= Surface tension

Step 2: Convert Radius to Meters

Given:

\)

Calculating:

\) $

Conclusion

The force required to overcome surface tension is 19.8 mN.

To find the correct equation for , we need to consider the forces acting on the liquid surface and how they relate to surface tension, gravity, and the curvature of the surface.

Surface tension acts to minimize the surface area of the liquid, leading to a curved surface. The curvature of the surface, in turn, produces a pressure difference described by the Young-Laplace equation:

where is the curvature of the surface. Since the angle is small, the equations governing the liquid surface simplify to linear terms.

The pressure difference caused by the liquid's weight at height is expressed as:

where is the density of the liquid and is the acceleration due to gravity.

Equating the pressure differences due to surface tension and gravity, we have:

Rearranging terms gives the differential equation for :

Thus, the correct equation describing the height of the liquid surface in the tank is: