Solutions (^171)
No separation occurs if the angle fl determined
from these equations is not smaller than a, and
hence
cos 13 < cos a.
Therefore, we find that the condition
v < gr (3 cos a — 2)
is a condition of crossing the boundary between
the planes by the wheels without separation. If
3 cos a — 2 < 0, i.e. a > arccos (2/3), the sepa-
ration will take place at any velocity v.
1.69. At the initial moment, the potential energy
of the system is the sum of the potential energy
mg (r + h) of the rim and the potential energy
pgh 2 I (2 sin a) of the part of the ribbon lying on the
inclined plane. The total energy of the system in
the final state will also be a purely potential
energy equal to the initial energy in view of the
absence of friction. The final energy is the sum of
the energy mgr of the rim and the energy of the
ribbon wound on it. The centre of mass of the latter
will be assumed to coincide with the centre of mass
of the rim. This assumption is justified if the
length of the wound ribbon is much larger than the
length of the circumference of the rim. Then the
potential energy of the wound ribbon is
p h
sin a
- s) gr,
the length of the ribbon being h/sin a + s, whore s
is the required distance traversed by the rim from
the foot of the inclined plane to the point at which
it comes to rest.
From the energy conservation law, we obtain
mg (r+h)+ pg
h 2
= + pgr.
2 sin a sin ce + s ,
whence
s —
nzg + p (h/sin a) (r—h/2)
pr