Variation of Pressure with Depth

One might guess that the deeper you go into a liquid or gas, the greater the pressure on you from the surrounding fluid will be. The reason for the increased pressure is that the deeper into a fluid you go, the more fluid, and thus the more weight, you have over top of you.

We can calculate the variation of pressure with depth by considering a volume of fluid of height h and cross-sectional area A (see Fig. 9.3).

  

Figure 9.3: Variation of Pressure with Depth

If this volume of fluid is to be in equilibrium, the net force acting on the volume must be zero. There are three external forces acting on this volume of fluid. These forces are:

1.

The force P_TA due to the pressure on top of the volume of fluid. If the fluid is open to the air, P_T = P_O = 1.01 x 10⁵ Pa, which is atmospheric pressure.

2.

The weight of the volume of fluid, w = Mg . Remembering the definition of density, $\rho$ = M/V , and that the volume of the fluid may be calculated as V = Ah , we can write the weight of the fluid as w = $\rho$ghA .

3.

The force pushing up on the bottom of the volume of fluid, P_BA , due to the fluid below the volume under consideration.

If we take the up direction to be positive and add the forces we get

P_BA - $$\rho$$ghA - P_TA = 0,

which gives

$$\rho$$ (9)

This provides the general formula relating the pressures at two different points in a fluid separated by a depth h .

Note: Only the density of the fluid and the difference in depth affects the pressure. The shape and size of the container are irrelevant. Thus the water pressure 6 inches below the surface of the ocean is the same as it is 6 inches below the the surface of a glass of salt water.

Idea: Pascal's Principle states that any pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid. Thus, in Fig.9.4,

  

Figure 9.4: Pascal's Principle

a pressure of P₁ = F₁/A₁ applied downward to the surface on the left of the container gets transmitted as an equal pressure upward of P₂ = P₁ on the surface on the other side of the container. The force on the other side is therefore:

$${A_2 \over A_1}$$ (10)

Thus if A₁ < A₂ , the transmitted force, F₂ , is greater than the applied force, F₁ . This is the principle behind the hydraulic press. For example, the transmitted force F₂ is used to balance the weight of a car in the hydraulic lift problem at the end of this chapter.