188 Aptitude Test Problems in Physicscase, AB = 21 cos 30° tan a, where 2/ is the side
of a hexagonal cross section of the pencil, and
2/ cos 30° is the radius of the circle inscribed in
the hexagonal cross section.
As long as point B lies in the hatched region,
the pencil will not roll down the plane.
Let us write the condition for the beginning of
rolling down
AD 1
=AB, or =1 jr5 tan a,
cos wo
cpo = arms ( _
1
- 1,/ 3 tan a )
Thus, if the angle p satisfies the conditionrtarccos _ (^) < stp <
1/3 tan a 2
the pencil remains in equilibrium. The expression
for the angle To is meaningful provided that
tan a > 1/ j/3. The fact that the pencil put paral-
lel to the generatrix rolls down indicates that
tan a > 1/ lig (prove this!).
Fig. 184
1.89. Let the cross section of the surface be de-
scribed by the function y (x) shown in Fig. 184.
Since the rod must be in equilibrium in any position,
cos To
whence