Problem 1.13 in lecture Notes
Compute the torque T required to rotate a conical object at a constant angular speed w. The clearance between the object and the casing is constant in thickness (h) and filled with oil of
 . (α = 30 o )
Solution: Assuming h is small enough to use a linearvelocity distribution within the clearance. Shear stress will develop on the conical surfaces of the object. The tangential velocity changes on this surface with radial distance. For simplicity let’s rotate the object 90 o .

D

2 Ro
 
 u
 y
,
  sin
 h
 wr

0 dr

Ro ds

____
AC
 wr h dF
  dA
  wr h
 rd
  ds dF
  wr
2 d
 
____
AC
Ro
 dr h dT up
T
UP

 r dF
0
Ro
 
2

0
 
 w
 h wr
3
____
AC h
Ro d
 
 r
3 d

____
AC
Ro
 dr dr 0 ≤ θ ≥ 2T
0 ≤ r ≥ R
T
UPPER
 
T
UP w
 h
AC
Ro

2
 
0
Ro r 3 dr
 
 w

Ro
3
2 h
____
AC
  w
 h
____
AC
Ro where
____
AC

D

Ro
4
4
, Ro


2

D
2
T
UP

T
TOTAL



T
 
 w

D
2
UPPER
3

D

2 h
T
LOW ER

T
TOTAL

2



  w D
4
16 h



 w D
4 
 
16 h
8 h

Due w D
4 to symetry they are equal
