projected-force-factor-cross-scan - Fluid Mechanics and Surface Tension Step by Step

A projected-force-factor cross-scan varies the contact angle across rows, showing how the cos(theta) projection scales the line force F = gamma * L * cos(theta). The arithmetic is exact only for the stated surface tension, line length, and angles in the ideal wetting model. Real measurements add wetting/contact-angle variation, contamination, curvature, temperature, instrument calibration, finite precision, and measurement uncertainty.

Projected contact force is scanned across bounded projection factors. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Projection starts after the line force is already checked

The projected pull does not choose its own source. It cites the two-surface line force and then applies a bounded rational factor.

F_{\parallel}=pF=\frac{1}{2}\cdot24\ \text{N}=12\ \text{N}

The projection factor gates how much pull remains

The line force is fixed in all three rows. The projected column changes only because the bounded factor changes.

\begin{array}{c|c|c}p&F&F_{\parallel}\\0&24\ \text{N}&0\ \text{N}\\\frac{1}{2}&24\ \text{N}&12\ \text{N}\\1&24\ \text{N}&24\ \text{N}\\\end{array}

The projected force cannot exceed its line-force source

The final diagram keeps the half projection tied to the line source. The table supplies the zero and full endpoints.

0\le F_{\parallel}\le F\qquad12\ \text{N}\le24\ \text{N}