Pascal’s Principle

Pascal’s principle (also known as Pascal’s law) states that when a change in pressure is applied to an enclosed fluid, it is transmitted undiminished to all portions of the fluid and to the walls of its container. In an enclosed fluid, since atoms of the fluid are free to move about, they transmit pressure to all parts of the fluid and to the walls of the container. Any change in pressure is transmitted undiminished.

Note that this principle does not say that the pressure is the same at all points of a fluid—which is not true, since the pressure in a fluid near Earth varies with height. Rather, this principle applies to the change in pressure. Suppose you place some water in a cylindrical container of height H and cross-sectional area A that has a movable piston of mass m (Figure 14.15). Adding weight Mg at the top of the piston increases the pressure at the top by Mg/A, since the additional weight also acts over area A of the lid:

Figure 14.15 Pressure in a fluid changes when the fluid is compressed. (a) The pressure at the top layer of the fluid is different from pressure at the bottom layer. (b) The increase in pressure by adding weight to the piston is the same everywhere, for example, $p_{\text{top new}}-p_{\text{top}}=p_{\text{bottom new}}-p_{\text{bottom}}$.

According to Pascal’s principle, the pressure at all points in the water changes by the same amount, Mg/A. Thus, the pressure at the bottom also increases by Mg/A. The pressure at the bottom of the container is equal to the sum of the atmospheric pressure, the pressure due to the fluid, and the pressure supplied by the mass. The change in pressure at the bottom of the container due to the mass is

Since the pressure changes are the same everywhere in the fluid, we no longer need subscripts to designate the pressure change for top or bottom:

Applications of Pascal’s Principle and Hydraulic Systems

Hydraulic systems are used to operate automotive brakes, hydraulic jacks, and numerous other mechanical systems (Figure 14.16).

Figure 14.16 A typical hydraulic system with two fluid-filled cylinders, capped with pistons and connected by a tube called a hydraulic line. A downward force ${\overrightarrow{F}}_1$ on the left piston creates a change in pressure that is transmitted undiminished to all parts of the enclosed fluid. This results in an upward force ${\overrightarrow{F}}_2$ on the right piston that is larger than ${\overrightarrow{F}}_1$ because the right piston has a larger surface area.

We can derive a relationship between the forces in this simple hydraulic system by applying Pascal’s principle. Note first that the two pistons in the system are at the same height, so there is no difference in pressure due to a difference in depth. The pressure due to $F_1$ acting on area $A_1$ is simply

According to Pascal’s principle, this pressure is transmitted undiminished throughout the fluid and to all walls of the container. Thus, a pressure $p_2$ is felt at the other piston that is equal to $p_1$. That is, $p_1=p_2.$ However, since$p_2=F_2\text{/}{A}_2,$ we see that

This equation relates the ratios of force to area in any hydraulic system, provided that the pistons are at the same vertical height and that friction in the system is negligible.

Hydraulic systems can increase or decrease the force applied to them. To make the force larger, the pressure is applied to a larger area. For example, if a 100-N force is applied to the left cylinder in Figure 14.16 and the right cylinder has an area five times greater, then the output force is 500 N. Hydraulic systems are analogous to simple levers, but they have the advantage that pressure can be sent through tortuously curved lines to several places at once.

The hydraulic jack is such a hydraulic system. A hydraulic jack is used to lift heavy loads, such as the ones used by auto mechanics to raise an automobile. It consists of an incompressible fluid in a U-tube fitted with a movable piston on each side. One side of the U-tube is narrower than the other. A small force applied over a small area can balance a much larger force on the other side over a larger area (Figure 14.17).

Figure 14.17 (a) A hydraulic jack operates by applying forces $(F_1\text{,}{F}_2)$ to an incompressible fluid in a U-tube, using a movable piston $(A_1,A_2)$ on each side of the tube. (b) Hydraulic jacks are commonly used by car mechanics to lift vehicles so that repairs and maintenance can be performed. (credit b: modification of work by Jane Whitney)

From Pascal’s principle, it can be shown that the force needed to lift the car is less than the weight of the car:

where $F_1$ is the force applied to lift the car, $A_1$ is the cross-sectional area of the smaller piston, $A_2$ is the cross sectional area of the larger piston, and $F_2$ is the weight of the car.

Example 14.3

Calculating Force on Wheel Cylinders: Pascal Puts on the Brakes

Consider the automobile hydraulic system shown in Figure 14.18. Suppose a force of 100 N is applied to the brake pedal, which acts on the pedal cylinder (acting as a “master” cylinder) through a lever. A force of 500 N is exerted on the pedal cylinder. Pressure created in the pedal cylinder is transmitted to the four wheel cylinders. The pedal cylinder has a diameter of 0.500 cm and each wheel cylinder has a diameter of 2.50 cm. Calculate the magnitude of the force $F_2$ created at each of the wheel cylinders.

Figure 14.18 Hydraulic brakes use Pascal’s principle. The driver pushes the brake pedal, exerting a force that is increased by the simple lever and again by the hydraulic system. Each of the identical wheel cylinders receives the same pressure and, therefore, creates the same force output $F_2$. The circular cross-sectional areas of the pedal and wheel cylinders are represented by $A_1$ and $A_2$, respectively.

Strategy

We are given the force $F_1$ applied to the pedal cylinder. The cross-sectional areas $A_1$ and $A_2$ can be calculated from their given diameters. Then we can use the following relationship to find the force $F_2$:

$$\frac{F_1}{A_1}=\frac{F_2}{A_2}.$$

Manipulate this algebraically to get $F_2$ on one side and substitute known values.

Solution

Pascal’s principle applied to hydraulic systems is given by $\frac{F_1}{A_1}=\frac{F_2}{A_2}:$

$$\begin{array}{}\\ \hfill F_2& =\frac{A_2}{A_1}{F}_1=\frac{\pi r_2^2}{\pi r_1^2}{F}_1\hfill \\ & =\frac{{\left(1.25\text{cm}\right)}^2}{{\left(0.250\text{cm}\right)}^2}\times 500\text{N}=1.25\times {\text{10}}^4\text{N}\text{.}\hfill \end{array}$$

Significance

This value is the force exerted by each of the four wheel cylinders. Note that we can add as many wheel cylinders as we wish. If each has a 2.50-cm diameter, each will exert $1.25\times {10}^4\text{N}\text{.}$ A simple hydraulic system, as an example of a simple machine, can increase force but cannot do more work than is done on it. Work is force times distance moved, and the wheel cylinder moves through a smaller distance than the pedal cylinder. Furthermore, the more wheels added, the smaller the distance each one moves. Many hydraulic systems—such as power brakes and those in bulldozers—have a motorized pump that actually does most of the work in the system.