It is sometimes the case that the effects of a force cannot be explained by the magnitude or direction of the force alone.  For example, let’s say that a friend of yours claims to be able to push a hole in a board solely with the force of his thumb.  You say that you don’t believe him, unless the board is of the “card” variety.  He produces the board and asks you to try pushing a hole in it with your thumb.  It’s a regular wooden board, and, of course, you can’t do it.  You tell your friend to go ahead and show you how it’s done.  He says OK, and proceeds to pull out a thumb tack.  You tell him that he’s cheating using the thumb tack, but he says that he is pushing on the board with no greater force than you did—the thumb tack is not helping him push any harder.  Indeed, he is only using the force of his thumb.  The difference, of course, is that the area over which the force was applied is significantly smaller with the thumb tack.  The force-per-unit-area is much, much greater.  (He also didn’t say anything about how big the hole was going to be!) 

A similar example can often be seen in old high school hallways.  If you look at a glancing angle down a linoleum hallway, you can often see that the flooring has many small indentations in it.  This is a result of many female teachers walking down the halls back in the 1940’s and 1950’s, not because of the weight of the females (although many high schools were know to hire Amazons back in the day...), but because of the fashions of the time—it was very much in vogue for women to wear stiletto heels that came to a sharp point at the bottom.  The force-per-unit-area on the floors was immense underneath the heels!

The force-per-unit-area mentioned in the discussion above is the pressure.

For a more formal definition of pressure, consider an object experiencing a force F applied to an area A of its surface.  We first find the component of the force F that is perpendicular to the area over which it is applied, denoted F_perp. 

The pressure (or, more precisely, the average pressure) exerted by the force F on the area A is then defined to be

Note that the MKS unit of pressure is the pascal, Pa.  In the problems we will be solving, the force F will usually already be perpendicular to the area A, so F_perp = F.

An important pressure with which we should be familiar is the atmospheric  pressure at the surface of the earth.  This pressure has the average value (at sea level)

You are probably already familiar with the fact that pressure increases as you  go down beneath the surface of a body of water.  This is in general true for any liquid (or gas, for that matter – the pressure at the earth’s surface is  certainly higher than it is at the top of the earth’s atmosphere!).

Consider a liquid having a density r_liq and a pressure p_top at its top surface.  The pressure p at a depth h beneath the liquid’s surface is then given by

p = p_top + rgh .