Transmission of forces in liquids

Pascal’s Principle applies when dealing with incompressible liquids. The particles in such substances exert forces on each other that do not preserve position but rather the average relative distance between them. If this did not occur, the particles would change their average relative distance in some regions, altering the total volume, indicating that we would be dealing with a compressible liquid. Most liquids are, in fact, highly incompressible, and therefore, when a force is applied in a given region, it is transmitted entirely throughout the fluid.

Consequences of Pascal’s Principle

The level is the same everywhere

A clear example of this is shown in the following figure. If this is filled with a liquid, the level will rise equally in all available spaces, regardless of the shape of the conduits.

This is because, as we have already seen, pressure is a function of height.

P = \rho g h

If there were a difference in height between different parts of the liquid, then there would necessarily be a difference in pressure between those parts.

Displaced Volume = Ceded Volume

If pressure is applied at some point in the liquid, it will displace while maintaining the same level in all other parts. The ceded volume v_1 would equal the displaced volume v_2+v_3.

Pascal’s Principle Applied in Hydraulic Machines

Pascal’s Principle is used in the construction of hydraulic presses. As pressure is distributed uniformly throughout all parts of a liquid, the following expression holds:

P_2 = P_1

But since pressure can be expressed as force per unit area, P=F/A, we have:

\displaystyle \frac{F_2}{A_2} = \frac{F_1}{A_1}

This means that if we apply a force F_1 on piston 1, a force F_2 will result on piston 2 as:

\displaystyle F_2 = \frac{A_2}{A_1} \cdot F_1

The factor A_2/A_1 is what we will call the “Amplification or reduction coefficient of the force.” It will amplify if it is greater than 1, and reduce if it is between 0 and 1.

Application Examples

1. Two people want to level a sloping terrain. To do this, they take a hose and fill it with water, then place its ends at a horizontal distance of 275[cm]. The water level at one point is 110[cm] and at the other is 175[cm].a) What is the difference in terrain level?

   b) What is the angle of inclination of the terrain?

   SOLUTION

2. A U-shaped hose has water inside, with a density of 1000 [kg/m^3]. If oil is poured into one end of the hose, filling it to a height of 20[cm], with a density of 800[kg/m^3], what level difference will be produced between the two ends of the hose?SOLUTION
3. Torricelli’s Barometer: Torricelli designed the following instrument to measure atmospheric pressure:a) If the density of mercury is 13.534,0[kg/m3] what will be the height of the mercury column due to atmospheric pressure? Consider P_{atm}=1,0[atm]=101.325,0[Pa]?

   b) If instead of mercury, water were used, how tall would the water column need to be to measure the same pressure?

   SOLUTION

4. A hydraulic lift has two pistons, one with a 2[foot] and the other with a 30[foot] radius. a) What weight must be placed on the smaller piston if a one-ton block is to be lifted on the larger piston? b) If the larger piston is to lift a concrete block by a height of 50[cm], how far must the smaller piston move?

   SOLUTION