Evangelista Torricelli was the first person to realize that...

Evangelista Torricelli was the first person to realize that we live at the bottom of an ocean of air. He correctly surmised that the pressure of our atmosphere is attributable to the weight of the air. The density of air at $0^{\circ} \mathrm{C}$ at the Earth's surface is $1.29 \mathrm{~kg} / \mathrm{m}^{3} .$ The density decreases with increasing altitude (as the atmosphere thins). On the other hand, if we assume the density is constant at $1.29 \mathrm{~kg} / \mathrm{m}^{3}$ up to some altitude $h$ and is zero above that altitude, then $h$ would represent the depth of the ocean of air. (a) Use this model to determine the value of $h$ that gives a pressure of 1.00 atm at the surface of the Earth. (b) Would the peak of Mount Everest rise above the surface of such an atmosphere?

Evangelista Torricelli was the first person to realize that we live at the bottom of an ocean of air. He correctly surmised that the pressure of our atmosphere is attributable to the weight of the air. The density of air at $0^{\circ} \mathrm{C}$ at the Earth's surface is $1.29 \mathrm{~kg} / \mathrm{m}^{3} .$ The density decreases with increasing altitude (as the atmosphere thins). On the other hand, if we assume the density is constant at $1.29 \mathrm{~kg} / \mathrm{m}^{3}$ up to some altitude $h$ and is zero above that altitude, then $h$ would represent the depth of the ocean of air.
(a) Use this model to determine the value of $h$ that gives a pressure of 1.00 atm at the surface of the Earth.
(b) Would the peak of Mount Everest rise above the surface of such an atmosphere?

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Key Concepts

Atmospheric Pressure

Atmospheric pressure is the force per unit area exerted by the weight of the air above a given surface. It is a fundamental concept in fluid mechanics and atmospheric science, illustrating how the mass of the air column creates pressure at the surface.

Hydrostatic Equilibrium

Hydrostatic equilibrium in a fluid indicates that the pressure at any point results from the weight of the fluid above that point. This principle leads to the relation p = ?gh in a simplified model, where p is the pressure, ? is the density, g is the acceleration due to gravity, and h is the height of the fluid column.

Simplified Atmospheric Model

The simplified atmospheric model assumes a constant density for air up to a specific altitude, beyond which the density abruptly drops to zero. This model allows a straightforward calculation of the effective height of the atmosphere by equating the weight of the air column (using p = ?gh) to the known atmospheric pressure.

Unit Conversion and Standard Values

Correct application of physical formulas requires consistency in units, such as converting the standard atmospheric pressure from atm to Pascals and using standard values for density and acceleration due to gravity. This ensures that calculations, like determining the effective height of the atmosphere, are accurate.

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