Question

A piston of cross-sectional area a is used in a hydraulic press to exert a small force of magnitude f on the enclosed liquid. A connecting pipe leads to a larger piston of cross-sectional area A (Figure). (a) What force magnitude F will the larger piston sustain without moving? (b) If the piston diameters are $\mathbf{3}\mathbf{.}\mathbf{80}\mathbf{}\mathbf{cm}$and $\mathbf{5}\mathbf{.}\mathbf{30}\mathbf{}\mathbf{cm}$, what force magnitude on the small piston will balance a $\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{kN}$ force on the large piston ?

Short answer

a) The force that the larger piston will sustain without moving is$\left(\begin{array}{l}\mathrm{A}\\ \mathrm{a}\end{array}\right)\mathrm{f}$

b) Magnitude of force is$103\mathrm{N}$

Step-by-step solution

The given data

(i) Diameter of small piston,$\mathrm{d}=3.8\mathrm{cm}\text{or}0.038\mathrm{m}$

(ii) Diameter of large piston,$\mathrm{D}=53\mathrm{cm}\text{or}0.53\mathrm{m}$

(iii) Force on larger piston,$\mathrm{F}=20\times {10}^3\mathrm{N}$

Understanding the concept of Pascal’s law

We can use Pascal’s law, which states that the change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and the wall of the container. In other words, we can say that the ratio of force to the area across the container would remain constant.

Formula:

Relation of pressure of larger and smaller piston using Pascal’s law,

$\frac{\mathrm{F}}{\mathrm{A}}=\frac{\mathrm{f}}{\mathrm{a}}$ (i)

Where,

$\mathrm{F}$and $\mathrm{A}$are the force and area of larger piston respectively; $\mathrm{f}$and$\mathrm{a}$are the force and area of the smaller piston respectively.

a) Calculation of force that will larger piston sustain

Using equation (i) and the given values, we get

$\mathrm{F}=\left(\begin{array}{c}\mathrm{A}\\ \mathrm{a}\end{array}\right)\mathrm{f}$

Hence, the equation of force that will define the force of larger piston is$\mathrm{F}=\left(\begin{array}{c}\mathrm{A}\\ \mathrm{a}\end{array}\right)\mathrm{f}$

b) Calculation of magnitude of force

Using equation (i) and the given values, we get the force on smaller piston as:

$\mathrm{F}=\left(\begin{array}{c}\mathrm{A}\\ \mathrm{a}\end{array}\right)\mathrm{f}$

$=\left(\frac{0{.038}^2}{0{.53}^2}\right)20\times {10}^3$

$=103\mathrm{N}$

Hence, the force required by smaller piston to balance the large piston is $103\mathrm{N}$