In this illustration we'll analyze a case
of a slit in a cylindrical vessel. we are
given that a wide cylindrical vessel of height
h is filled with a liquid of density ro, on
the vessel wall their is a narrow slit running
down to the bottom of vessel, and the slit
length is l, and its width is equal to b.
and it is saying when liquid is filled in
a vessel slit is closed if suddenly the slit
is opened we are required to find the force
of reaction of water flowing out, just after
opening the slit. here to understand this
situation lets first draw the, figure, on
ground we are given that a wide cylindrical
vessel is kept. on this ground, say if this
is a wide, cylindrical vessel is kept over
here. and in this vessel, we are made a slit,
on it side walls say this is a rectangular
slit. which is made. and we are given that
the slit length is l. and width of slit is
equal to b, that means this width. is given
as b. so this is a rectangular slit, narrow
slit is made in the side wall, and, liquid
is completely filled in it. which is water
in this case. and, in this situation the vessel
height is also given to us as h. and water
density here we can, consider as ro, which
is also given to us. now when suddenly the
slit is opened water will start coming out
from the slit throughout, this length. so
due to the ejection of water, the vessel will
experience a backward reaction force. which
we can calculate, by considering an element
from the top, at a depth x, bellow the, open
end of the vessel, and we consider an element
of slit of width d x. so here we can say if
we consider. an, elemental, opening, in slit.
of area. this will be b d x because the width
is b and, length we are considering as d x.
then, we can say, if flux velocity. of water.
from this opening, is. this flux velocity
we can say as it is considered at a depth
x bellow the open end. so this will be root
2 gee x. and we can write the force of reaction.
due to. water ejection. from this element.
is, this we can write as d f and the value
we know the reaction force is given as ro
ay v square we already studied in concepts.
so here we can it write ro, area is given
as b d x for this elemental opening. and the
velocity is, root 2 gee x, whole square. so
the value of, total reaction force, we can
give as integration of d f which is, here,
we can take the constants out, which is 2
gee ro b. and this is integration of x d x.
and limits of integration here we substitute
from this point to this point. that will be
from, h minus l to. h. and if we integrate
this the value of force i continue here. this
will be given as 2 gee, ro, b, multiplied
by its integration is. h square x square by
2, and we apply limits from, h minus l to
l. now you can simplify after substitution
of limits and this will give us the value.
this 2 gets cancelled out and the final result
will be getting is, ro, g, l, b, multiplied
by 2 h, minus l, that will be the result of
this problem.
