Why don't we feel that much atmospheric pressure?

Question

I tried to lift an umbrella upwards by its handle with its concave face facing upwards. If the atmosphere exerts a 1Bar pressure and the radius of the umbrella is 30 cm, then the downward force on it should have been 9000$\pi$N after neglecting the weight of the umbrella, and the reaction force of air and assuming that its surface is circular. However, the force required to lift it hardly felt like lifting 900$\pi$Kg. The force required was very less.

Also, if we consider our body to be a cylinder of radius 15cm, then the extra downward force on our body due to atmospheric pressure should be 2250$\pi$N which means a tremendous increase in normal reaction and hence apparent weight. However, that clearly doesn't happen.

Answer 1

There is air underneath the umbrella as well as above, not as much as above it, but it feels the pressure of the atmosphere around it. The atmospheric pressure affects that air under the umbrella, giving it the same pressure as the air, so the total pressure on the umbrella is balanced out.

If you put a ruler under a large sheet of paper on the top of a table, with a short piece of the ruler sticking out beyond the paper, and then press downward on it, you will feel the atmosphere air pressure as a force making it difficult for you to push the ruler down.

But as soon as the paper lifts up, even by a small amount, then the atmospheric air pressure will equalise and you will find it easy to lift the paper up.

Also, if we consider our body to be a cylinder of radius 15cm, then the extra downward force on our body due to atmospheric pressure should be $2250ππN$ which means a tremendous increase in normal reaction and hence apparent weight. However, that clearly doesn't happen.

Actually you could extend this question to "Why am I not being crushed by the pressure above?"

If you watch any TV program about aquatic life at great depths, the "problem" of being flattened is even worse for them.

But in their case, their bodies are basically bags of fluid at the same pressure as the water around them. So the forces on the bodies of these creatures are equalized.

In the human case, air inside your body, pushing outward at the same pressure, allows for equalisation.

EDIT This answer is an inadvertent duplication of: Human Bodies and Atmospheric Pressure END EDIT

Answer 2

Note that the pressure acts in all directions and not only in one direction. An object with surface $S$ can be subdivided into very small surfaces $\Delta S$. On every point of this surface a unit normal vector $\vec{n}$ can be assigned. If a pressure $p$ acts on a point $x$ in the surface, one has the force:

$\Delta \vec{F} = p \vec{n} \Delta S$.

Adding all forces on the surface together leads to a resulting force. In the special case of the umbrella, the surface can be (little simlified) seen as decomposed in two region: One region where unit normal vector points to the top (upper site of the umbrella) and another region where unit normal vector points downward. Adding both components together and assuming that pressure on upper and lower site is nearly equal (umbrella is very thin and pressure function is continuous punction), forces on upper site and lower site nearly cancel.

Answer 3

Why dont you also consider that there is air pressure under the umbrella too. The two forces cancel and you only have the apply a force equal to that objects weight.