Let’s study about torricelli's theorem.
this theorem is about, calculation of, efflux
velocity, of a fluid, flowing out, through
a hole, in a container. in many applications
we need to use it. and, its analytical treatment
is done on the basis of bernoullie’s theorem
only. say we consider a situation, in which
we are given with, a cylindrical container,
and it is placed on ground. and say the container
is filled with a liquid of density ro, and
at the top it is open to atmosphere, as atmospheric
pressure is applied at the top. and say, from
the topmost level of liquid, at a depth h
below, we make a hole in the container, through
which the liquid starts coming out. the velocity
with which it comes out we call it efflux
velocity, and this efflux velocity can be
calculated, by using bernoullie’s theorem,
at 2 points, 1 is the point a, at which the
fluid is coming out and other point we can
take, a point b which is inside, or we can
also apply it at a point b which is at the
top of the level. now in this situation, if
we use, bernoullie’s theorem, at just inside,
and outside points, of hole, see what we’ll
get, just outside we can take pressure to
be p atmospheric, and if the efflux velocity
is v, the ki-netic energy, of the fluid flowing
out, can be taken as, half ro v square. and,
we can ignore gravitational potential energy
as both of these points are at the same horizontal
level, this must be, the pressure at point
b which can be taken as, p atmospheric plus,
h ro g, plus, the ki-netic energy here can
be ignored, it can be taken as zero as the
area of cross section of container is considered
to be very large compared to that of the hole.
or we can simply write, area of hole, to be
very very less than, the area of container.
so we can ignore the velocity with which the
fluid is flowing inside the container. and
here p atmospheric also gets cancelled out,
and here ro will also cancel out, then the
efflux velocity can be written as, root 2
g h. and this equation we call, torricelli's
equation, of efflux velocity. and, this equation
we will use in different cases. soon we are
going to see, a situation when area of hole
is not very small, in that case efflux velocity
will also depend on area of hole, but, if
we are given that area of hole is very small
compared to cross sectional area of container,
then efflux velocity is given by this expression.
it is only dependent on, the height, of free
surface of liquid upto which it is filled,
above the point from where efflux, is coming
out.
