Have you ever been to a Science exhibition? Have you witnessed one or two projects that utilize water to operate it? In Science projects like a Hydraulic crane, students utilize the incompressible property of water to make its components work. When a student pushes a syringe inwards, the water reaches the other end, and the crane functions. Did you know that it is secretly an application of Pascal's law?
Pascal's Law
A French physicist Blaise Pascal proposed a law to explain the transmission of pressure in liquid. It states, “Liquid in an enclosed container exerts pressure equally in all directions.” It is popularly known as Pascal's Law for Liquid Pressure.
Mathematically,
\[\rm P_{A} = P_{B} = P_{C} = \dots = P_{D}\]
, where $\rm P_{i}$ denotes the pressure at points A, B, C, …, and N, respectively.
Exploiting the Pascal's Law
As long as liquid remains in a closed container, Pascal's law holds. It states that water pushes each side of the closed container with equal thrust per unit area.
Let A and B be two syringes connected via a tube filled with water (liquid). As there are no openings through this system, we can treat the system as a closed container. If we connect one of the syringes by pushing the plunger inwards and the other by pulling the plunger outwards, we can understand Pascal's law better.
When we push the plunger of syringe A inwards, we apply some force to the plunger, say $\rm F_{A}$. Then the water from
$$\rm P_{A} = P_{B}$$
$$\rm or, \frac{F_{A}}{A_{A}} = \frac{F_{B}}{A_{B}}$$
$$\rm or, F_{A} = \frac{ A_{A} }{ A_{B} } \cdot F_{B}$$
$$\rm or, F_{A} = k \cdot F_{B}$$