Solutions (^179)
The total distance traversed by the block rela-
tive to the conveyer belt is
, 21 , v 2 _ (v-I-Jr2tter
s s 2 = / v I (^) pg 1- 214
2ttg (^) •
The amount of heat liberated at the expense of
the work done by friction is
m (v
1.79. In the former case (the motion of the pipe
without slipping), the initial amount of potential
energy stored in the gravitational field ill be
transformed into the kinetic energy of the pipe,
which will be equally distributed between the
energies of rotary and translatory motion. In the
latter case (the motion with slipping), not all the
potential energy will be converted into the kinet-
ic energy at the end of the path because of the
work done against friction. Since in this case the
energy will also be equally distributed between the
energies of translatory and rotary motions, the
velocity of the pipe at the end of the path will be
smaller in the latter case.
1.80. After the spring has been released, it is uni-
formly stretched. In the process, very fast vibra-
tions of the spring emerge, which also attenuate
very soon. During this time, the load cannot be
noticeably displaced, i.e. if the middle of the
spring has been displaced by a distance x in doing
the work A, the entire spring is now stretched by
x. Therefore, the potential energy of the spring,
which is equal to the maximum kinetic energy in the
subsequent vibratory motion, is Wk= kx 2 /2, where
k is the rigidity of the entire spring. When the
spring is pulled downwards at the midpoint, only
its upper half (whose rigidity is 2k) is stretched,
and the work equal to the potential energy of ex-
tension of the upper part of the spring is A =
2k (x 2 /2) = kx 2. Hence we may conclude that the
maximum kinetic energy of the load in the subse-
quent motion is Wk = A/2.
= p,mgs —
2 •
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