Bukhovtsev-et-al-Problems-in-Elementary-Physics

240 ANSWERS AND SOLUTIONS
The closer the sections of the rod are to the axis of rotation, the more
wi11 they be stretched.
204. In the reading system that is stationary with respect to the axis, the
force of tension of the rod does not perform any work, since this force constantly
remains 'perpendicular to the velocity of the ball. In the moving system
this force performs work other than zero, and this changes the kinetic energy
of the ball.
205. Section AB of the hoop with a mass m at the highest point has an
energy of mg 2R+m ~V)2. During motion the kinetic and potential energies
of section AB begin to decrease owing to the work of the forces of the elastic
deformation of the hoop whose resultant produces a centripetal force always
directed towards the centre. The velocity of section AB forms an obtuse angle a
with the force F (Fig. 366). For this reason the work of the force W 1=F!1S cos a
is negative, and, consequently, the energy of the section with the mass m
diminishes.
After section AB passes through the lowermost position, it is easy to see
that the work of the force F becomes positive and the energy of section AB
will grow.
206. Let us draw a tangent to the inner circumference of the spool (Fig. 367)
from point A, which is the instantaneous axis of rotation (see Problem 173).
If the directions of the thread and the
tangent AC coincide, the moment
of the forces that rotate the spool around the instantaneous axis will be zero.
Therefore, if the spool is at rest, it will not rotate around the instantaneous
axis and, consequently, it will not move translationally.
The angle a at which the motion of the spool is reversed can be found from
triangle AOB; namely, sin a = ; . If the thread is inclined more than a,
the spool will roll to the right, If a Is smaller it will move to the left on
condition that it does not slip. If the tension of the thread T satisfies the
condition Tr ez.iR, where f is the force of friction, the spool will remain in
place. Otherwise, when sin a= ~ it will begin to rotate counterclockwise
around point o.
207. Break the hoop into equal small sections each with a mass of L\m.
Consider two symmetrical sections (relative to the centre). All the particles of
the hoop simultaneously participate in translational motion with the velocity
Fig. 366 Fig. 367