Pascal's Principle — Question 5

Question 5

Two output pistons need not start moving together

Two vertical output pistons share a hydraulic supply at the same height. Piston A has area $10\,\mathrm{cm^2}$ and total downward load $200\,\mathrm{N}$; B has area $20\,\mathrm{cm^2}$ and load $600\,\mathrm{N}$. Both backs are atmospheric. They begin on lower stops. Ignore friction and weight beyond the stated loads.

1. As supply gauge pressure rises slowly, determine which piston lifts first and whether both can be freely stationary at one common pressure.

2. If A reaches an upper stop before pressure rises further, find its downward stop reaction when B is just about to lift.

Question 5 - Solution

Two output pistons need not start moving together

(a) Derivation and calculation

At incipient lift the lower-stop reaction is zero:$$\boxed{p_A=L_A/A_A=200/.00100=200\,\mathrm{kPa}}.$$$$\boxed{p_B=L_B/A_B=600/.00200=300\,\mathrm{kPa}}.$$$$p_A<p_B\Rightarrow\boxed{A\ \text{lifts first}}.$$Both cannot freely balance at one pressure because $L_A/A_A\ne L_B/A_B$.

(b) Result and interpretation

When B is just about to lift, $p_g=300\,\mathrm{kPa}$. For A at its upper stop,$$p_gA_A-L_A-R_A=0.$$$$\boxed{R_A=(300000)(.00100)-200=100\,\mathrm N\ \text{downward}}.$$B has zero lower-stop reaction at that threshold.

Physical check

Stops supply the additional reactions that allow unequal load-per-area pistons to share a common pressure.