Answer to Question #143108 in Math for Usman

Question

Determine the intensity of shear of an oil having viscosity = 1.2 poise and is used for lubrication in the clearance between a 10 cm diameter shaft and its journal bearing. The clearance is 1.0 mm and shaft rotates at 200 r.p.m.

Answer 1

Given:

$\mu=1.2poise=0.12N\cdot s/m^2$

Diameter of shaft $D=10cm=0.1m$

Distance between shaft and journal bearing $dy=1.0mm=10^{-3}m$

Speed of shaft $N=200\ r.p.m.$

Tangential speed of shaft is given by

$$u=\dfrac{\pi DN}{60}=\dfrac{\pi\cdot0.1m\cdot200\ r.p.m.}{60}=\dfrac{\pi}{3}m/s$$

Change of velocity between shaft and bearing

$$du=u-0=\dfrac{\pi}{3}m/s$$

Mathematically

$$\tau=\mu\dfrac{du}{dy}=0.12N\cdot s/m^2\times\dfrac{\dfrac{\pi}{3}m/s}{10^{-3}m}=$$

$$=40\pi N/m^2\approx125.6637 N/m^2$$