let us discuss about the practical situation
of, cross section of a freely falling liquid.
say if we draw situation like, say we are
given with a tap, and from this tap, if water
starts flowing, we know well that, as liquid
falls, the area of cross section decreases,
and it happens because of, continuity equation,
the rate at which the liquid is coming through
the tap, with the same rate it has to flow
down. so in this situation the area of cross
section of, the flowing fluid has to decrease.
say if, ay is the area of cross section at
the tap, and at a depth h below, the area
of cross section is taken as a. in this situation
we can say, if the fluid is coming through
the tap with velocity v not, and at a depth
h it is moving with a speed v, then we can
find out the relation between v and v not
by using bernoullie’s theorem. say this
point is x and this point is y. here we can
use, bernoullie’s theorem, at points, x
and y, this’ll give us, at x we can write
pressure to be p atmospheric, plus ki-netic
energy is, half ro v not square, plus, it
is h height above the level at y, at this
point it must be having, a potential energy
ro g h higher per unit volume compared to
point y. and at point y we can write pressure
to b p atmospheric and ki-netic energy to
be, half ro v square. and as already we have
taken zero reference level at y, so here p
atmospheric gets cancelled out, and the density
of liquid also cancels. so the relation we
are getting is, v square is v not square,
plus 2 g h, which is just the speed equation
in free fall because, liquid is falling freely.
so here bernoullie’s theorem is just verifying
the laws of gravity also. so in this situation
if we apply the continuity equation, we can
write, using equation of continuity, here
we can write, v not ay is equal to, v-a. as
the flow rate at section x is equal to, flow
rate of water at y. so here cross sectional
area at depth h can be given as, v not by
v into a. and if we substitute the value of
speed from here, the cross sectional area
at depth h can be given as v not ay by, root
of v not square plus 2 g h. this is also very
useful result, you must keep on your tips.
or whenever required, this is just an application
of bernoullie’s theorem by which we are
getting it using continuity equation also.
so just, keep this analysis in mind, because
there are various numerical applications similar
to this situation also where we need to apply
this.
