Figure 9.21 (a) shows a thin liquid film supporting a small weight = 4.5 x 10^-2 N. What is the weight supported by a film of the same liquid at the same temperature in Fig. (b) and (c)? Explain your answer physically. ![Fig 9.21 -- Three panels of a thin liquid film on the same 40 cm wire frame with a sliding bar and weight, each panel drawn with a different overall film shape](/figures/mechanical-properties-of-fluids/liquid-film-wire-frame-shapes-fig-9-21.svg)

Figure 9.21 (a) shows a thin liquid film supporting a small weight = 4.5 x 10^-2 N. What is the weight support

Exercises · 9.18

Q. Figure 9.21

(a) shows a thin liquid film supporting a small weight = 4.5 x 10^-2 N. What is the weight supported by a film of the same liquid at the same temperature in Fig.

(b) and (c)? Explain your answer physically.

Fig 9.21 -- Three panels of a thin liquid film on the same 40 cm wire frame with a sliding bar and weight, each panel drawn with a different overall film shape — Figure 9.21

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Concept understanding — Capillary Action

Capillary Action: The Physics of Water Defying Gravity

You've seen it happen a hundred times. Dip the corner of a paper towel into a spill, and watch the water climb upward into the towel — against gravity. Or take a thin glass tube (a capillary), put it in water, and the water rises inside it, higher than the outside surface. That's capillary action.

The name comes from capillus, Latin for "hair" — because the effect is strongest in tubes as thin as a hair.

The Intuition: Two Forces at War

Think of water molecules as tiny magnets. They attract each other strongly (cohesion), and they also stick to the walls of a container (adhesion). In a narrow tube, adhesion to the glass pulls the water upward along the walls. The water molecules, clinging to each other, drag the entire column up with them.

But gravity pulls down. The water rises only until the upward pull from adhesion is exactly balanced by the weight of the water column. That's the equilibrium height.

Note

If adhesion is stronger than cohesion (water-glass), the liquid rises. If cohesion is stronger (mercury-glass), the liquid is depressed — it falls below the outside level. Mercury doesn't wet glass.

The Precise Physics: What Determines the Height?

For a liquid that wets the tube (contact angle $\theta < 90^\circ$ ), the rise height $h$ is given by the Jurin's law:

h = \frac{2\gamma \cos\theta}{\rho g r}

Where:

$\gamma$ = surface tension of the liquid (N/m)
$\theta$ = contact angle between liquid and tube wall
$\rho$ = density of the liquid (kg/m³)
$g$ = acceleration due to gravity (9.8 m/s²)
$r$ = radius of the tube (m)

h = \frac{2\gamma \cos\theta}{\rho g r}

The key insight: $h$ is inversely proportional to $r$ . Halve the tube radius, and the water rises twice as high. That's why the effect is only noticeable in very narrow tubes — in a wide pipe, $h$ is negligible.

Why Does Surface Tension Pull Upward?

The surface tension $\gamma$ acts along the circumference of the water-air interface inside the tube. The total upward force is:

F_{\text{up}} = (2\pi r) \times \gamma \cos\theta

The weight of the water column (height $h$ , density $\rho$ ) is:

F_{\text{down}} = (\pi r^2 h) \times \rho g

Set them equal, cancel $\pi r$ , and you get Jurin's law.

Tip

For water in clean glass, $\theta \approx 0^\circ$ (perfect wetting), so $\cos\theta = 1$ . Then $h \approx \frac{2\gamma}{\rho g r}$ . For water at room temperature, $\gamma \approx 0.073$ N/m, so $h \approx \frac{0.015}{r}$ (with $r$ in metres). A tube of radius 0.1 mm gives a rise of about 15 cm.

Real-World Examples

Paper towels and sponges: The fibres form millions of tiny capillary channels. Water rises through them, soaking the towel. …

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The weight a liquid film can support equals the surface-tension force on its sliding wire, $F = 2SL$ , which depends only on the surface tension $S$ and the length $L$ of the wire touching the film — not on the film's area, height or shape. All three arrangements share the same liquid, temperature and 40 cm wire, so all three support the same weight, $4.5 \times 10^{-2}$ N.

Concept

Surface tension $S$ is a force per unit length acting along any line in the surface of the film. A free liquid film has two surfaces (a front and a back), so along the sliding wire of length $L$ the film pulls up with a total force

F = 2 S L.

This force balances the supported weight $W$ .

Why shape and area do not matter

The pulling force acts only along the line where the film meets the movable wire; it is set by the length $L$ of that contact line, not by how tall or wide the film is above it. Enlarging, shrinking or reshaping the film above the wire changes its area but leaves $L$ unchanged, so the supported weight is unchanged. (Also, an equilibrium film has uniform tension, so the extra triangular flap in (c) adds no extra pull on the wire.)

Steps

From arrangement (a): $W_a = 2 S L$ with $W_a = 4.5 \times 10^{-2}$ N and $L = 0.40$ m. …

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Q.Figure 9.21

Concept

Why shape and area do not matter

Steps

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