Pascal's Principle — Question 8

Question 8

Seal friction changes both the threshold and the energy balance

A hydraulic device has input area $2.0\,\mathrm{cm^2}$ and output area $20\,\mathrm{cm^2}$. During slow upward output motion, the output seal exerts $100\,\mathrm{N}$ downward friction in addition to a $900\,\mathrm{N}$ load. Input friction is negligible. Both backs are atmospheric and height differences are neglected.

1. Find the required input force. For a $0.020\,\mathrm{m}$ output rise, find the input travel.

2. Calculate input work, useful load work and dissipated energy. Compare with the frictionless force ratio.

Question 8 - Solution

Seal friction changes both the threshold and the energy balance

(a) Derivation and calculation

For slow upward output motion,$$p_gA_2=L+F_f=900+100=1000\,\mathrm N.$$$$p_g=1000/.00200=500000\,\mathrm{Pa},\quad\boxed{F_1=p_gA_1=100\,\mathrm N}.$$$$\boxed{x_1=(A_2/A_1)x_2=10(.0200)=.200\,\mathrm m}.$$

(b) Result and interpretation

$$\boxed{W_{\mathrm{in}}=100(.200)=20.0\,\mathrm J},\quad\boxed{W_{\mathrm{useful}}=900(.0200)=18.0\,\mathrm J}.$$$$\boxed{E_{\mathrm{dissipated}}=100(.0200)=2.00\,\mathrm J}.$$$$L/F_1=9<10=A_2/A_1,\qquad W_{\mathrm{in}}=W_{\mathrm{useful}}+E_{\mathrm{dissipated}}.$$

Physical check

Pressure transmission remains valid while the moving seal dissipates mechanical energy.