/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 48 An air bubble of radius \(r\) in... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 48

An air bubble of radius \(r\) in water is at a depth \(h\) below the water surface at some instant. If \(P\) is atmospheric pressure and \(d\) and \(T\) are the density and surface tension of water, respectively, the pressure inside the bubble will be (A) \(P+h d g-\frac{4 T}{r}\) (B) \(P+h d g+\frac{2 T}{r}\) (C) \(P+h d g-\frac{2 T}{r}\) (D) \(P+h d g+\frac{4 T}{r}\)

Short Answer

Expert verified
The pressure inside the air bubble is given by \(P_{inside} = P + h d g + \frac{2T}{r}\), which corresponds to option (B).

Step by step solution

01

Write down the given information

We are given the following information: - Air bubble radius: \(r\) - Depth below the water surface: \(h\) - Atmospheric pressure: \(P\) - Density of water: \(d\) - Surface tension of water: \(T\)
02

Calculate the hydrostatic pressure at depth h

To find the hydrostatic pressure at the depth \(h\), we can use the formula: \(P_h = P + h d g\) where - \(P_h\) is the hydrostatic pressure at depth \(h\) - \(P\) is the atmospheric pressure - \(h\) is the depth - \(d\) is the density of water - \(g\) is the acceleration due to gravity
03

Account for the additional pressure due to surface tension

For the additional pressure due to surface tension, we can use the Young-Laplace equation for a sphere: \(\Delta P = \frac{2T}{r}\) where - \(\Delta P\) is the additional pressure due to surface tension - \(T\) is the surface tension of water - \(r\) is the radius of the air bubble
04

Combine the hydrostatic pressure and additional pressure to find the pressure inside the bubble

Now, to find the pressure inside the bubble, we can simply add the hydrostatic pressure and the additional pressure due to surface tension: \(P_{inside} = P_h + \Delta P\) Substituting the expressions from Steps 2 and 3 gives us: \(P_{inside} = (P + h d g) + \frac{2T}{r}\)
05

Check the answer against the given options

Comparing the expression for the pressure inside the bubble with the given options, we see that it matches option (B): \(P_{inside} = P + h d g + \frac{2T}{r}\) So, the correct answer is option (B).

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