Pascal's Principle — Question 9

Question 9

A lever and a hydraulic stage multiply different quantities

A lever drives a master piston. The applied force acts $0.30\,\mathrm{m}$ from the pivot, and the piston rod acts $0.060\,\mathrm{m}$ from it; both forces are perpendicular to the lever. The master area is $2.0\,\mathrm{cm^2}$ and the output area $30\,\mathrm{cm^2}$. Neglect all losses and elevation effects.

1. Find the output force from an applied lever force of $40\,\mathrm{N}$.

2. For a small lever-end travel of $15\,\mathrm{mm}$, find the output travel and verify work conservation to first order in the lever rotation.

Question 9 - Solution

A lever and a hydraulic stage multiply different quantities

(a) Derivation and calculation

$$F_{\mathrm{in}}(.300)=F_m(.0600)\Rightarrow\boxed{F_m=200\,\mathrm N}.$$$$p_g=200/.000200=1.00\,\mathrm{MPa}.$$$$\boxed{F_{\mathrm{out}}=p_gA_{out}=(10^6)(.00300)=3000\,\mathrm N}.$$$$F_{\mathrm{out}}/F_{\mathrm{in}}=5(15)=75.$$

(b) Result and interpretation

For a small common lever rotation,$$x_m=x_{\mathrm{in}}(.060/.300)=15/5=3.00\,\mathrm{mm}.$$$$\boxed{x_{\mathrm{out}}=x_mA_m/A_{\mathrm{out}}=3.00/15=.200\,\mathrm{mm}}.$$$$\boxed{W_{\mathrm{in}}=40(.0150)=.600\,\mathrm J=3000(.000200)=W_{\mathrm{out}}}.$$

Physical check

Displacement ratios invert the force multipliers, preserving work to first order in the lever rotation.