d by the symbol $T$ and has units of force per unit length.

Work done in forming a bubble: When you blow a bubble, you increase its surface area. This requires work to be done against the surface tension forces trying to minimize the area.

Relationship Between Work, Surface Area, and Surface Tension

The work done (W) in increasing the surface area of a liquid is:

$$W = T \Delta A$$

where:

$T$ is the surface tension

$\Delta A$ is the change in surface area

Bubble 1:

Radius = $R$

Surface Area = $4\pi R^2$ (Since a soap bubble has two surfaces, inner and outer)

Work Done = $W_1$

Bubble 2:

Radius = $2R$

Surface Area = $4\pi (2R)^2 = 16\pi R^2$

Work Done = $W_2$

Temperature's Effect:

Surface tension of liquids generally decreases with an increase in temperature.

Calculations

Since the surface area of bubble 2 is four times that of bubble 1, you might think $W_2 = 4W_1$. However, the surface tension is lower for bubble 2 due to the heated soap solution.

Conclusion

Because the surface tension decreases with temperature:

Work done ($W_2$) to form bubble 2 will be less than four times the work ($W_1$) needed to form bubble 1.

Therefore, the correct answer is C: $W_2 < 4W_1$

Fluid Mechanics

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