## Illustration 14.1: Pressure in a Liquid

With fluids, instead of discussing forces, we usually talk about pressure, which is defined as the force per unit area or P = F/A. This is because the direction of the force a liquid exerts on its container depends on the shape of the container (force is normal to the surface of the container) and the size of the container. Pressure is not a vector (no direction) and does not depend on the size of the container (position is given in meters and pressure is given in pascals). Restart.

Move the pressure indicator in the tube and note the pressure readings (the pressure is only measuring the effect of the liquid asdescribed below). Let's discuss why pressure increases as a function of depth. Assume the blue liquid is water (density 1000 kg/m3). Pick a point to measure the pressure somewhere in the upper tube. If the dimension of the container into the screen (the dimension you cannot see) is 1 m, what is the volume of water above the point you picked? What is the mass and thus the weight of the water at that point? For example, consider a depth of 3 m. The pressure is 29,400 N/m2. The volume of water above this point is a cylinder of volume 9.4 m3. The mass of the water is the volume times the water's density, or 9,400 kg, and therefore the weight of the water is 92,120 N.

What is the force downward at that point? Well take the weight and divide by the cross-sectional area of the column of water at that point, which is 3.14 m2. This pressure should be equal to the pressure reading. The units of pressure are N/m2 = pascals (abbreviated Pa).

Strictly speaking, this is the gauge pressure, not the absolute pressure, because we assumed P = 0 at the top of the water column when the pressure (due to the atmosphere) is actually around 1 x 105 Pa. The absolute pressure then would be the pressure at the top due to the atmosphere added to the pressure due to the weight of the water. All of this comes together in the equation:

P = P0 + ρgy,

where P0 is the pressure at the top, ρ is the density of the liquid, g is acceleration due to gravity and y is the depth of the liquid.

What will be the pressure at point A? Add a second pressure indicator to check.

Illustration authored by Anne J. Cox.

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