Pascal's Principle — Question 6

Question 6

A small pump must deliver enough volume as well as pressure

A hydraulic pump piston of area $1.0\,\mathrm{cm^2}$ makes delivery strokes of length $0.10\,\mathrm{m}$. Ideal check valves refill it on each return stroke. A lift piston of area $80\,\mathrm{cm^2}$ must rise $0.050\,\mathrm{m}$ against a constant $1600\,\mathrm{N}$ load. Ignore height-head changes and losses.

1. Find the number of complete delivery strokes required and the force during a delivery stroke.

2. Find total input work and explain why check valves are essential to accumulating lift over repeated strokes.

Question 6 - Solution

A small pump must deliver enough volume as well as pressure

(a) Derivation and calculation

$$\Delta V_{\mathrm{stroke}}=A_px_p=(10^{-4})(.100)=10^{-5}\,\mathrm{m^3}.$$$$V_{\mathrm{lift}}=A_Lx_L=.00800(.0500)=4.00\times10^{-4}\,\mathrm{m^3}.$$$$\boxed{N=V_{\mathrm{lift}}/\Delta V_{\mathrm{stroke}}=40}.$$$$p_g=1600/.00800=200000\,\mathrm{Pa},\quad\boxed{F_p=p_gA_p=20.0\,\mathrm N}.$$

(b) Result and interpretation

$$\boxed{W_{\mathrm{in}}=NF_px_p=40(20.0)(.100)=80.0\,\mathrm J}.$$$$W_{\mathrm{out}}=1600(.0500)=80.0\,\mathrm J.$$The outlet valve prevents backflow during refill, while the inlet admits replacement liquid. Without this sequence, a return stroke would tend to reverse the delivered volume.

Physical check

Forty full strokes supply both the required volume and the required work.