Pascal's Principle — Question 1

Question 1

Force amplification is paid for with input travel

An ideal hydraulic device has piston areas $A_1=2.0\,\mathrm{cm^2}$ and $A_2=50\,\mathrm{cm^2}$. A downward input force $F_1=80\,\mathrm{N}$ acts on the small piston. Both backs are exposed to the same atmosphere. Neglect piston weight, friction, elevation differences and liquid compression.

1. Find the gauge pressure and supported output force.

2. If the small piston moves down $0.15\,\mathrm{m}$, find the output rise and compare input and output work.

Question 1 - Solution

Force amplification is paid for with input travel

(a) Derivation and calculation

$$A_1=2.0\times10^{-4},\quad A_2=5.0\times10^{-3}\,\mathrm{m^2}.$$$$\boxed{p_g=F_1/A_1=80/(2.0\times10^{-4})=400\,\mathrm{kPa}}.$$$$\boxed{F_2=p_gA_2=(400000)(.0050)=2000\,\mathrm N},\quad F_2/F_1=25.$$

(b) Result and interpretation

$$A_1x_1=A_2x_2\Rightarrow\boxed{x_2=.150/25=.00600\,\mathrm m}.$$$$\boxed{W_1=80(.150)=12.0\,\mathrm J},\quad\boxed{W_2=2000(.00600)=12.0\,\mathrm J}.$$Force amplification is balanced by reduced output travel.

Physical check

The common atmospheric pressure cancels from both piston balances, leaving gauge pressure.