The tube has an inner diameter of d=2 mm and is immersed in water. Determine the average length

step 1 Okay, so the surface tension of water sigma is 75 .4 x 10 -3 Nm Angle of content of water with t

The tube has an inner diameter of d=2 mm and is immersed in...

The tube has an inner diameter of $d=2 \mathrm{~mm}$ and is immersed in water. Determine the average length $L$ to which the water will rise along the tube due to capillary action as a function of the angle of tilt, $\theta$. Plot this relationship of $L$ (vertical axis) versus $\theta$ for $10^{\circ} \leq \theta \leq 30^{\circ} .$ Give values for increments of $\Delta \theta=5^{\circ} .$ The surface tension of the water is $\sigma=75.4 \mathrm{mN} / \mathrm{m},$ and its density is $\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}$.

Key Concepts

Tube Geometry and Projection Effects

The geometry of the tube, including the inner diameter, is crucial because it determines the radius of curvature of the liquid meniscus and directly influences the magnitude of capillary rise. When the tube is tilted, the effective vertical height achieved by the liquid is modified by a projection factor (typically involving the cosine of the tilt angle), linking the actual length of liquid along the tube to its vertical component.

Hydrostatic Equilibrium

Hydrostatic equilibrium describes the balance between the upward capillary forces and the downward gravitational forces acting on the column of liquid. In capillary rise, the weight of the liquid column (which increases with height) is balanced by the pressure generated by surface tension, establishing the equilibrium height of the liquid in the tube.

Capillary Action

Capillary action is the phenomenon where a liquid spontaneously rises or falls in a narrow tube due to the competition between adhesive forces (between the liquid and the tube material) and cohesive forces (within the liquid itself). This interplay results in the bending of the liquid’s free surface and the rise (or depression) of the liquid in the tube, a critical concept underlying the behavior of fluids in confined spaces.

Surface Tension

Surface tension is a measure of the cohesive forces between liquid molecules at the interface with another medium (such as air). It plays a pivotal role in capillary action by creating a pressure difference across the curved surface of the liquid, which then drives the liquid upward against gravity in narrow tubes.

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