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Consider a 1.80-m-tall man standing vertically in water and completely submerged in a pool. Determine the difference between the pressures acting at the head and at the toes of the man, in kPa. Take the density of water to be 1000 kg/m3.

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klefenz

Answer:

17.658 kPa

Explanation:

The hydrostatic pressure of a fluid is the weight of a column of that fluid divided by the base of that column.

[tex]P = \frac{weight}{base}[/tex]

Also, the weight of a column is its volume multiplied by it's density and the acceleration of gravity:

[tex]weight = \delta * v * g[/tex]

Meanwhile, the volume of a column is the area of the base multiplied by the height:

[tex]V = base * h[/tex]

Replacing:

[tex]P = \frac{\delta * base * h * g}{base}[/tex]

The base cancels out, so:

[tex]P = \delta * h * g[/tex]

The pressure depends only on the height of the fluid column, the density of the fluid and the gravity.

If you have two point at different heights (or depths in the case of objects submerged in water) each point will have its own column of fluid exerting pressure on it. Since the density of the fluid and the acceleration of gravity are the same for both points (in the case of hydrostatics density is about constant for all points, it is not the case in the atmosphere), we can write:

[tex]\Delta P = \rho * g * (h1 - h2)[/tex]

We do not know at what depth the man of this problem is, but it doesn't matter, because we know the difference in height of the two points of interes (h1 - h2) = 1.8 m. So:

[tex]\Delta P = 9.81 \frac{m}{s^{2} } * 1000 \frac{kg}{m^3} * 1.8 m = 17658 Pa = 17.658 kPa[/tex]

xero099

The difference between the pressures acting at the head and at the toes of the man is 17.563 kilopascals.

In this case, the 1.80-m-tall man experiments an hydrostatic pressure, whose difference between the head and toes is determined by the concept of manometric pressure:

[tex]\Delta P =\frac {\rho \cdot g \cdot h} {1000}[/tex] (1)

Where:

  • [tex]\rho[/tex] - Fluid density, in kilograms per cubic meter.
  • [tex]g[/tex] - Gravitational acceleration, in meters per square second.
  • [tex]h[/tex] - Height difference between head and toes, in meters.
  • [tex]P[/tex] - Pressure, in kilopascals.

If we know that [tex]\rho = 1000\,\frac{kg}{m^{3}}[/tex], [tex]g = 9.807\,\frac{m}{s^{2}}[/tex] and [tex]h = 1.80\,m[/tex], then the pressure experimented by the 1.80-m-tall man is:

[tex]\Delta P = \frac{\left(1000\,\frac{kg}{m^{3}} \right)\cdot \left(9.807\,\frac{m}{s^{2}} \right)\cdot (1.80\,m)}{1000}[/tex]

[tex]\Delta P = 17.653\,kPa[/tex]

The difference between the pressures acting at the head and at the toes of the man is 17.563 kilopascals.

We kindly invite to check this question on hydrostatic pressures: https://brainly.com/question/19201184

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