Pascal's Principle — Question 2

Question 2

Equal pressure increments do not erase an elevation difference

Two hydraulic pistons are connected by liquid of density $900\,\mathrm{kg/m^3}$. The output piston is $1.2\,\mathrm{m}$ higher than the input piston. Their areas are $A_1=3.0\,\mathrm{cm^2}$ and $A_2=60\,\mathrm{cm^2}$. Equal atmospheric pressure acts on their backs. Ignore weights and friction and use $g=10\,\mathrm{m/s^2}$.

1. What input force supports a $1200\,\mathrm{N}$ output load at this fixed geometry?

2. If the input force increases by $6.0\,\mathrm{N}$ while both pistons are held at their original heights, how much does the output support force increase?

Question 2 - Solution

Equal pressure increments do not erase an elevation difference

(a) Derivation and calculation

$$p_{g,2}=L/A_2=1200/.00600=200000\,\mathrm{Pa}.$$$$p_{g,1}=p_{g,2}+\rho g\Delta z=200000+900(10)(1.20)=210800\,\mathrm{Pa}.$$$$\boxed{F_1=A_1p_{g,1}=(.000300)(210800)=63.24\,\mathrm N}.$$

(b) Result and interpretation

At fixed piston elevations, the hydrostatic offset stays constant:$$\Delta p_2=\Delta p_1=\Delta F_1/A_1=6.0/.000300=20000\,\mathrm{Pa}.$$$$\boxed{\Delta F_2=A_2\Delta p_2=.00600(20000)=120\,\mathrm N}.$$Pascal transmission concerns equal pressure increments, not equal absolute pressures at different heights.

Physical check

Ignoring the elevation term would underestimate the initial input force by 3.24 N.