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Calculating Gauge Pressure and Depth: Damage to the Eardrum · Physics Playground

Exercises Fluids

Example 11.13 from College Physics, 11.9 Pressures in the Body

Calculating Gauge Pressure and Depth: Damage to the Eardrum

Suppose a 3.00-N force can rupture an eardrum. (a) If the eardrum has an area of $1.00 {cm}^{2}$ , calculate the maximum tolerable gauge pressure on the eardrum in newtons per meter squared and convert it to millimeters of mercury. (b) At what depth in freshwater would this person’s eardrum rupture, assuming the gauge pressure in the middle ear is zero?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy for (a)
The pressure can be found directly from its definition since we know the force and area. We are looking for the gauge pressure.
Solution for (a)
$P_{g} = F / A = 3 . 00 N / (1 . 00 \times 10^{- 4} m^{2}) = 3 . 00 \times 10^{4} {N/m}^{2} .$
11.57

We now need to convert this to units of mm Hg:

$P_{g} = 3 . 0 \times 10^{4} {N/m}^{2} (\frac{1.0 mm Hg}{133 {N/m}^{2}}) = 226 mm Hg.$
11.58
Strategy for (b)
Here we will use the fact that the water pressure varies linearly with depth $h$ below the surface.
Solution for (b)
$P = hρg$ and therefore $h = P / ρg$ . Using the value above for $P$ , we have

$h = \frac{3.0 \times 10^{4} {N/m}^{2}}{(1.00 \times 10^{3} {kg/m}^{3}) (9.80 m {/s}^{2})} = 3.06 m.$
11.59
Discussion
Similarly, increased pressure exerted upon the eardrum from the middle ear can arise when an infection causes a fluid buildup.

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From College Physics (1st edition) by OpenStax (Rice University), section 11.9, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.

Strategy for (a)

Solution for (a)

Strategy for (b)

Solution for (b)

Discussion