**Pressure energy in a
liquid at rest**

We have
in a roundabout way come to recognise pressure energy but we have done it for a
liquid that is flowing steadily. Now we must decide whether the idea can be
extended to liquid that is at rest. Figure 3-6 shows a tank filled with
liquid that is at rest. An element of liquid having volume and having the shape of a thin disc forms
part of the free surface. The depth of liquid in the tank is and so the potential energy of the element
relative to the bottom of the tank is where *r* is the
density of the liquid.

The element could be moved very slowly to a new position at a depth of and, as the movement involves no net force outside the liquid, no work would be done during the movement. This means that the element has lost potential energy equal to without doing any work nor apparently gaining any other form of mechanical energy in exchange. As this statement is contrary to the law of the conservation of energy we must again face the need to look for, and try to recognise, a form of mechanical energy other than potential energy and kinetic energy. As the only observable change in the water in the element is the rise in its pressure, we must expect this new form of energy to be associated with pressure.

The absolute pressure at the
surface is the atmospheric pressure and so, for the element when it is on the surface,
the product of pressure and volume is *.* At depth the absolute pressure is and the product of pressure and volume is *.*
Clearly, for the element, the product of pressure and volume has increased by which is numerically equal to the loss of
potential energy. It is also equal to the mass of the element times the same
expression as we used to define pressure energy/unit mass. So it seems that the
concept of pressure energy, which has been devised to permit us to quantify the
energy entering or leaving a closed system, can also be used to quantify the
energy in store in a liquid at rest. However it is still only true because the
liquid is continuous and able to flow in the gravitational field.

When the element is at depth its potential energy relative to the datum is . Our interest then lies in the fact that the sum of this potential energy and the pressure energy possessed by the element is:-

and that this is the same
wherever the element may be in the tank. We can deduce that in a liquid at rest
the sum of the potential energy/unit weight and the pressure energy/unit weight
at every point is the same and equal to *H*.
These two forms of energy are interchangeable.