In this illustration, we'll analyze the case
of squeezing water out from a tube. here the
figure shows a cylindrical tube which has
an orifice of area small s through the water.
contained in the tube is pushed out, by applying
a constant force on piston, in time t. if
initial volume of water in tube is v we are
required to find the work done by the force
in squeezing, all the water out from tube.
so here we can directly write if, length of
tube, is l. then, we can directly write work
done by, force, is, this work done we can
directly write as f l. because when, the force
will squeeze, whole of the liquid, out.
then we can say the total length l will, give
us a work done f into l. now in this situation,
if, we use, bernoulli's theorem. just inside.
and outside.
the orifice. here we can see, in this case
just inside the orifice we can say, the pressure
will be here it is, p atmospheric plus, the
force applied by, external agent is, giving
us, an excess pressure f, upon s. so in this
situation we can write, the pressure inside
the orifice is p not plus, f by, s. which
is the pressure just inside due to external
atmospheric pressure plus due to the external
force, and we can neglect the fluid velocity
inside.
and just outside the orifice we can write
p not is the atmospheric, pressure.
plus the kinetic energy of fluid is half ro,
v square.
here p not gets cancelled out. and we get
the value of force applied in terms of, the
flux velocity is half.
ro v square, s. and, in this situation.
we can write if, discharge velocity. is, v.
then in this situation we can write, in time t.
whole volume.
of water, v, will come out.
this implies we can write, the total volume
is equal to volume flow rate which is small
s multiplied by discharged velocity multiplied
by time, so flow rate is area of orifice multiplied
by the flux velocity we multiplied with, time,
will give us the total volume which is squeezing
out.
so this is giving us the flow velocity. of,
liquid which is coming out that is v divided
by s t.
so in this situation here we are getting the
value of, force.
so, force, magnitude.
here we are getting is equal to f is equal
to here, half, ro, v square we substitute
here as v upon, s t.
whole square.
multiplied by the area of cross section of
the tube, so in this situation if we calculate
the work done, total work done.
then this total work done we have seen it
is given as f l. so we multiply this force
with l, see what we are getting half ro, v
square by, s square t square multiplied by
s, l. here capital s we are taking the area
of cross section, of this whole tube. here
you can also write this s is, area of, cross
section.
of tube, so this s multiplied by l again can
be written as the total volume of tube so
this can be written as. v cube ro, divided
by, twice of, s square, t square that is the
total work done, in squeezing the whole, water
out of the tube. which is the result of this
problem.
