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

MECHANICS 187
Fig. 296
..,-----........
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/' ,
/ A '.
I \
I \
\
IJJ
'......
o
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I
(71'
When the sacks are exchanged simultaneously,
the final velocities of the
boats v~ and 0; can be found from the
equations:
Mvo-mvo=(M +m) v~ ;
-Mvo+mvo=(M·+m) v~
, , M-m
Hence. VI =-V2 = M +m Vo' Thus, the
final velocity of the boats will be higher
in the first case.
85. External forces do not act in a horizontal
direction on the "hoop-beetle" systern.
For this reason the centre of gravity
of the system (point C in Fig. 296) will not
move in a horizontal plane. The distance
from the centre of gravi ty of the syst em
.. to the centre of the hoop is CO=m;M R.
Since this distance is constant, the centre of the hoop 0 will describe a circle
with the radius CO about the stationary point C. It is easy to see t hat the tra ..
jectory of the beetle is a circle with the radius AC=m~M R.
The mutual positions and the direction of motion of the beetle and the
hoop are shown in Fig. 296.
86. Since no external forces act on the system in a horizontal direction.
the projection of the total momentum of the "wedge-weights" system onto the
horizontal direction must. remain constant (equal to zero). It thus follows that
the wedge will begin to move only if the weights move.
For the weight m 2 to move to the right, the condition
should be observed.
Therefore. ml E;; sin a-k cos a. Here the wedge will move to the left. For
the weight m~to move to the leit.ithe following condition should be observed:
or
mlg~ m~ sin ex+km~cos ex
Here the wedge will move to the right.
Hence, for the wedge to be in equillbrtum,
of the weights should satisfy the inequality
the ratio between the masses
sin a.-k cos a~ ml ~ sin a+k cos a.
mj