Question

Figure 9.21 (a) shows a thin liquid film supporting a small weight = 4.5 × 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.

Answer 1

Given (Figure 9.21 (a))

• Weight supported by the film in (a): \( W_a = 4.5 \times 10^{-2} \,\text{N} \)
• Length of the film supporting the weight in each case: \( l = 40\,\text{cm} = 0.4\,\text{m} \)
• Same liquid and same temperature in all three figures (a), (b), (c).

Surface tension from case (a)

A liquid film has two free surfaces, so the total length along which surface tension acts is \(2l\).

For case (a): Surface tension \( T = \dfrac{W_a}{2l} = \dfrac{4.5 \times 10^{-2}}{2 \times 0.4} = 5.625 \times 10^{-2}\,\text{N m}^{-1} \)

Weight in (b) and (c)

The liquid and temperature are the same in (a), (b), and (c), so the surface tension \(T\) is the same in all three cases.

In (b) and (c), the length of the film that supports the weight is again \( l = 0.4\,\text{m} \), and the film still has two free surfaces. Hence, the supporting force is again \( F = 2Tl \), which is unchanged from case (a).

Therefore, the weight supported in (b) and in (c) is the same as in (a): \( W_b = W_c = 4.5 \times 10^{-2}\,\text{N} \).

Physical explanation

• The film supports the weight because surface tension pulls along the edges of the film; for a film, this force is \( F = 2Tl \) (two surfaces, each of length \(l\)).
• Since both \(T\) (same liquid and temperature) and \(l\) (same length) are the same in (a), (b), and (c), the maximum weight the film can balance must be the same in all three figures.