Pascal’s law & hydraulic machine

PASCAL’S LAW:

Pascal’s law can be stated as:

“The pressure applied externally at any point of a liquid enclosed in a container is transmitted equally to all parts of the liquid in container”.

• In general, this law holds good for fluids, both for liquids as well as gases.

EXPREIMENTAL PROOF:

Consider a water filled glass vessel having holes of uniform width around its surface and an opening fitted with moveable rigid piston. When force $F$ is applied through the piston, it exerts pressure on water. This pressure is transmitted equally throughout the liquid in all directions and the water rushes out of all holes with the same pressure.

HYDRAULIC MACHINE:

A hydraulic machine is a device that uses pressurized fluids to transfer energy and perform mechanical tasks. These machines use the principles of Pascal's law to amplify force and transmit power. Common examples of hydraulic machines include hydraulic brakes, cranes, car lifts and hydraulic jacks.

WORKING:

Consider a hydraulic press or lift consist of two pistons which are connected by an incompressible fluid filled pipe, as shown in figure:

A force of magnitude $F_1$ is applied to the piston of small surface area $A_1$. The pressure $P_1$produce by first piston is then transmitted uniformly through the fluid. Same amount of pressure $P_1$ acts on the second piston of larger surface area $A_2$. The larger area of second piston cause the farce of larger magnitude act on it and this piston can easily displace upward. Hence, just be increasing the area of piston we can amplify force as per the principle of Pascal’s law.

MATHEMATICAL EXPRESSION:

$\text{Pressure at Piston 1:}$

$F_1 = {P_1}{A_1}$

OR

$\boxed{P_1 =\frac{F_1}{A_1}}$ ………. (i)

$\text{Pressure at Piston 2:}$

$F_2=P_1A_2$

Since, the pressure is same across the two pistons, i.e. $P_1$.

We can write,

$\boxed{P_1 =\frac{F_2}{A_2}}$ ………. (ii)

Compare equation (i) and (ii):

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

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

OR

$\boxed{{F_2} =F_1 \text {x}\frac{ A_2}{A_1}}$

CONCLUSION:

From the above equation we conclude that the force $F_2$ is greater than the force $F_1$ by a factor of $\frac{A_2}{A_1}$. By designing a hydraulic press with appropriate areas $A_1$ and $A_2$ , a large output force can be obtained by exerting the small input force.