33
Chapter 1 x Introduction
Separating the variables, we may integrate:
w
³
Zo
dZ
Z
SPro4
5SP ro2 t º
»
¬ 3mh sinT ¼
ª
t
2hIo sinT ³0
dt, or: Z
Z o exp «
1.54* A disk of radius R rotates at
angular velocity : inside an oil container
of viscosity P, as in Fig. P1.54. Assuming a
linear velocity profile and neglecting shear
on the outer disk edges, derive an expression for the viscous torque on the disk.
Ans.
Fig. P1.54
Solution: At any r d R, the viscous shear W | P:r/h on both sides of the disk. Thus,
d(torque) dM 2rW dA w
or:
P1.55
M 4S
P:
h
R
³
r 3 dr
0
2r
P:r
h
2Sr dr,
SP:R4
h
Ans
A block of weight W is being pulled
W
over a table by another weight Wo, as shown in
h
Fig. P1.55. Find an algebraic formula for the
steady velocity U of the block if it slides on an
oil film of thickness h and viscosity P. The block
bottom area A is in contact with the oil.
Neglect the cord weight and the pulley friction.
Wo
Fig. P1.55
Solution: This problem is a lot easier to solve than to set up and sketch. For steady motion, there
is no acceleration, and the falling weight balances the viscous resistance of the oil film: