Question

Where do you observe Pascal’s principle in our daily life? Give a few examples.

Answer 1

Hint: Recall that Pascal’s principle suggests an equal and undiminished transmission of pressure throughout all points of an enclosed incompressible fluid. Also, we know that pressure is inversely proportional to the area upon which a force is incident, and hydrostatic pressure is proportional to the density, the acceleration due to gravity and the height of a fluid column. To this end, determine instances where we aim to obtain a large force by applying a small force over a small area, or the effect of increasing depth on atmospheric/ hydrostatic pressure.

Formula used:
Complete step by step answer:
Let us begin by understanding what Pascal’s Principle entails.
Pascal’s Principle (also known as the principle of transmission of fluid-pressure) suggests that the pressure applied at any point of liquid enclosed in a container is transmitted undiminished to all other parts of the liquid in the container. This means that any change in pressure at any point in a confined incompressible fluid is transmitted throughout the fluid such that the same change occurs everywhere.
Pascal’s law is given as $P=\dfrac{F}{A}$, where P is the pressure exerted, and A is the area upon which the force F is incident.
Change in hydrostatic pressure is given as: $\Delta P = \rho g \Delta h$, where $\rho$ is the density of the liquid, g is the acceleration due to gravity and h is the height of the fluid column. The intuitive interpretation of this relation is that the difference in pressure between two elevations of fluid in a column is due to the weight of the fluid in between the two elevations.
Let us now look as some examples of Pascal’s law in daily life:
1. Hydraulic lift:
A hydraulic lift consists of two cylinders, a narrow cylinder A connected to a wider cylinder B, both filled with an incompressible fluid, and are fitted with airtight pistons. One piston is where we apply a force and the other piston houses the load that needs to be lifted.
So, when a pressure equivalent to the pressure needed to lift the load to a certain height is applied on the smaller piston, this pressure is transmitted throughout all direction in the liquid, and since the liquid is common to both the cylinders, this pressure gets exerted on the piston of the second cylinder which lifts the load upwards with a greater force.

Usually, the force piston/cylinder is narrow whereas the load piston/cylinder is wide to facilitate exerting a force over a smaller area which produces a larger pressure with less effort. However, the price we pay is we have to push the piston further down to obtain more pressure over the smaller area, which pushes the load piston up by a distance less than the distance we pushed the force piston down, since the pressure gets distributed over a larger area of the load piston.
Thus, since $P_A = P_B$ and $A_2$ > $A_1$ and $P=\dfrac{F}{A}$, then we have $F_2$> $F_1$

2. Scuba diving:
As scuba divers dive deep into the ocean, they experience different oceanic pressure at different depths of water. This is the result of hydrostatic pressure building up upon layers of water with increasing depth of the ocean. Every layer of water exerts hydrostatic pressure in all directions. This means that deeper the dive, the more water there is above the diver, and the more pressure that is exerted on the diver as a result of transmission of the cumulative weight of the water layers throughout all layers of the water in the ocean. This is why scuba divers experience more additive pressure as they dive deeper.
3. Hydraulic brakes:
Most automobiles use hydraulic brakes that work on the principle of Pascal’s law. To apply the brakes, the foot pedal is pushed due to which a pressure is exerted via the piston, on the liquid of the master cylinder which in turn pushes the liquid from the master cylinder to the wheel cylinder. As a consequence of Pascal’s principle, the pressure exerted is transmitted equally and undiminished throughout the liquid of the wheel cylinder and the pistons $B_1$ and $B_2$ get pushed outwards with equal pressures.

As a result, the brake shoes get pressed against the rim of the wheel due to which the wheel can no longer execute any motion and brings the vehicle to a halt. The pressure from the master cylinder is thus transmitted to all the wheels of the vehicle which retards the motion of every wheel due to the undiminished transmission of pressure throughout the fluid in the pipeline connecting all the wheels.

Note: Remember that a hydraulic lift can also be used as a hydraulic press, by substituting the load with an object that needs to be compressed and by enclosing the area over the load piston by a barrier that serves to compress the object. By applying a force on the narrow piston, we are able to obtain a larger force on the wider piston which pushes the object up and the barrier consequently compresses the object. Thus all hydraulic systems working this way are also known as force multipliers, since the resultant force exerted on the load will be much greater than the applied force.