SCIENCE · INVESTIGATE AND EXPLAIN

Use contact geometry to predict capillary rise or depression

Cohesion refers to attraction between molecules of the same material; adhesion refers to attraction across different materials. Surface tension γ is force per length, with units N/m. It describes the tendency of an interface to reduce area. It is not the same quantity as pressure, although a curved interface can sustain a pressure difference.

Learning outcome: Compare contact angles 0°, 90° and 120° at two radii. Retain signed height and signed vertical surface force, and explain why using the absolute value of cosine would destroy the physical distinction.

Starting recap

Declare density, pressure reference, flow regime and admissible force constraints. Convert area and volume units before balancing forces, mass or energy.

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Try

At radius 0.5 mm and contact angle 0°, the capillary rise is 0.024 m. At the same radius and 120° the depression is −0.012 m.

Open the experiment

Predict

For the stated angle convention, a contact angle above 90° predicts:

Your prediction and reason

Experiment

Compare contact angles 0°, 90° and 120° at two radii. Retain signed height and signed vertical surface force, and explain why using the absolute value of cosine would destroy the physical distinction.

Resolve the sign and geometry of capillary displacement

Worked teaching model · synthetic values · Ideal static cylindrical capillary, surface tension .06 N/m, liquid density 1000 kg/m³, g=10. Contact angle is measured through liquid. Signed height relative to a broad reservoir; ignore meniscus volume and molecular-scale corrections. Virtual only.

Tube radius: 1 mm Contact angle through liquid: 60 degrees

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Rise or depression versus contact angle

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: ideal capillary balance (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Rise or depression versus contact angle

ideal capillary balance

contact angle (degrees)	signed height (m)
0	0.012
3	0.0119836
6	0.0119343
9	0.0118523
12	0.0117378
15	0.0115911
18	0.0114127
21	0.011203
24	0.0109625
27	0.0106921
30	0.0103923
33	0.010064
36	0.0097082
39	0.00932575
42	0.00891774
45	0.00848528
48	0.00802957
51	0.00755184
54	0.00705342
57	0.00653567
60	0.006
63	0.00544789
66	0.00488084
69	0.00430042
72	0.0037082
75	0.00310583
78	0.00249494
81	0.00187721
84	0.00125434
87	0.000628031
90	0
93	-0.000628031
96	-0.00125434
99	-0.00187721
102	-0.00249494
105	-0.00310583
108	-0.0037082
111	-0.00430042
114	-0.00488084
117	-0.00544789
120	-0.006

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Signed capillary height	0.006	m
Vertical surface-tension resultant	0.000188496	N
Signed hydrostatic column force	0.000188496	N
Contact-angle cosine	0.5	ratio

Wetting geometry sets the sign; the narrower tube magnifies the signed capillary displacement.

Balance 2 pi r gamma cos(theta)=rho g pi r² h. Contact angle below 90 degrees gives rise, above it gives depression. Cohesion describes attraction within a liquid and adhesion across different materials. Surface tension has units N/m, not pressure; meniscus curvature can nevertheless produce a pressure difference.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Wetting geometry sets the sign; the narrower tube magnifies the signed capillary displacement.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Declare the fluid, reference and evidence

Build the algebraic model

In the ideal cylindrical capillary, contact angle θ is measured through the liquid. The vertical component of the surface force around circumference 2πr is 2πrγcosθ. Balancing the signed hydrostatic column force ρgπr²h yields h=2γcosθ/(ρgr). This derivation neglects meniscus volume and assumes a broad external reservoir.

Check the original numerical packet

With γ=0.06 N/m, ρ=1000 kg/m³, g=10, radius 1 mm and θ=60°, height is +0.006 m. At 90° the vertical component and displacement are zero. At 120° the cosine is negative and the model predicts depression −0.006 m. Halving radius doubles the magnitude at the same angle; it does not change the sign.

Challenge the assumptions and transfer

A separate curved-interface calculation distinguishes surfaces. For a supplied spherical liquid drop, Δp=2γ/R; a thin soap film enclosing gas has two surfaces and Δp=4γ/R. At γ=0.03 N/m and R=0.002 m, the respective differences are 30 and 60 Pa. Doubling radius halves them. These ideal geometry packets do not instruct blowing bubbles, handling liquids or measuring biological pressures.

Worked original case

At radius 0.5 mm and contact angle 0°, the capillary rise is 0.024 m. At the same radius and 120° the depression is −0.012 m.

Bounded investigation

Changed-case numerical practice

Try the changed case before feedback

With γ=0.05 N/m, ρ=1000 kg/m³, g=10, r=1 mm and contact angle 0°, find capillary rise in millimetres.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. All records are supplied or generated educational evidence. No pressure rigs, liquid handling, bodily measurements or physical exposure protocols are requested. Virtual admissibility does not certify real equipment safety, practical skill or independent mastery.

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Public practice checks a specific claim. It does not establish independent mastery.

Why does an attraction-based microscopic explanation still require a specified macroscopic contact angle?

Your explanation and limitations

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