An air bubble, 1 cm in diameter, is released from the regulator of a scuba diver swimming 30 m below the sea surface. Estimate the diameter of the bubble just before it reaches the water surface.
This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
This question is designed to encourage you to think of a body of water as a source of a pressure gradient, so that things that move around in that body of water can experience different pressures as it moves. Then it asks to consider a bubble rising.
In practice, bubbles that rise are deformed from their static circular shape, but calculating that is a much more advanced problem, so presuming you are a student you are implicitly being encouraged to ignore that detail and assume that the bubble is spherical for simplicity.
We assume the bubble remains spherical and that the air behaves as an ideal gas, so Boyle’s law applies. In this case
p_{30}V_{30}=p_0V_0\,,
and for a sphere the volume is proportional to the diameter cubed, so
p_{30}d_{30}^3=p_0d_0^3\,.
Thus,
d_0=d_{30}\sqrt[3]{\frac{p_{30}}{p_0}}\,,
where
- at 30 m depth the pressure is
p_{30}=p_\mathrm{atm}+\rho\,g\,(30\;\mathrm{m})\approx101325~\mathrm{Pa}+1000~\mathrm{kg/m^3}\times9.8~\mathrm{m/s^2}\times30~\mathrm{m}\approx395325~\mathrm{Pa}\,,
- at the surface the pressure is
p_0\approx101325~\mathrm{Pa}\,.
With d_{30}=1~\mathrm{cm} the diameter at the surface is
d_0=1~\mathrm{cm}\sqrt[3]{\frac{395325}{101325}}\approx1~\mathrm{cm}\sqrt[3]{3.9}\,.
The cube root of 3.9 is approximately 1.57 so
d_0\approx1.57~\mathrm{cm}\,.
Thus, the bubble’s diameter just before reaching the water surface is approximately 1.57 cm.
To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 3.4: Hydrostatic Distributions in Atmosphere and Ocean (HIGHLY RELEVANT)
- Chapter 11.5: Surface Tension (HIGHLY RELEVANT)
- Chapter 4.3: Archimedes’ Law (HIGHLY RELEVANT)
- Chapter 3 How Stuff Works: How a Boyle depth gauge works (RELEVANT)
- Chapter 3 Engineering Models: Hydrostatic Equation (RELEVANT)
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