Torricelli's Theorem :
We will learn that
The speed of liquid coming out through a hole at a depth h below the free surface is the same as that of a particle fallen freely through the height h under gravity. This is known as Torricelli’s theorem. The speed of the liquid coming out is called the speed of efflux.
Speed Of Efflux :
The word ‘efflux’ means the outflow of the fluid. Let us find an expression for the velocity of efflux for a fluid, from small hole of its container. Consider a closed vessel filled with a liquid up to height
l
. A small hole (or orifice) is made in its side at a depth
h
below the surface of the liquid
s
shown in Figure.
Taking the liquid to be incompressible and its flow through the hole as streamlined, we can apply the equation of continuity at points 1 and 2
A
2
v
2
=
A
1
v
1
Applying Bernoulli’s equation at the two points, we have
If the cross-sectional area of the vessel
A
2
is much larger than that of the orifice i.e.,
A
1
then,
v
2
= 0
Since the hole is open to atmosphere so, the pressure
P
1
is the same as the atmospheric pressure
P
a
.
…(1)
This expression gives the velocity of efflux.
Case 1:
Torricelli's Law:
In case the vessel containing the fluid is open i.e., not covered, the pressure
P
a
at the top of the liquid surface is same as the atmosphere pressure
P
a
. Thus, the velocity of efflux given above in equation (i) becomes
…(2)
This is same as the speed acquired by a body after falling freely through a height
h
. The expression (2) is known as Torricelli’s law.
Case 2:
In case pressure
P
2
>>
P
a
, we can ignore the value 2
gh
in equation (1), to get the speed of efflux as
The speed of efflux is determined by the container pressure. This occurs during rocket propulsion.
Suppose, the surface of a liquid in a tank is at a height h from the orifice
O
on its sides, through which the liquid issues out with velocity
v
. The speed of the liquid coming out is called the speed of efflux. If the dimensions of the tank be sufficiently large, the velocity of the liquid at its surface may be taken to be zero. Applying Bernoulli’s equation at the surface and just outside the orifice.
With
h
= 0 at the orifice, we have
Evangelista Torricelli
show that this velocity is the same as the liquid will attain in falling freely through the vertical height (
h
) from the surface to the orifice. This is known as
Torricelli's theorem
and may be stated as, "
The velocity of efflux of a liquid issuing out of an orifice is the same as it would attain if allowed to fall freely through the vertical height between the liquid surface and orifice
."
Range [R]
Let us find the range
R
on the ground.
Considering the vertical motion of the liquid,
Now, considering the horizontal motion,
R
=
vt
or
or
From the expression of
R
, following conclusions can be drawn,
(i)
This is shown in fig.
(ii)
R
is maximum at
and
Proof
R
2
= 4(
Hh
–
h
2
)
For
R
to be maximum,
That is,
R
is maximum at
And
Hence Proved.
Time Taken To Empty A Tank
We are here interested in finding the time required to empty a tank if a hole is made at the bottom of the tank.
Consider a tank filled with a liquid of density ρ upto a height
H
. A small hole of area of cross section a is made at the bottom of the tank. The area of cross-section of the tank is
A
.
Let at some instant of time the level of liquid in the tank is
y
. Velocity of efflux at this instant of time would be,
Now, at this instant volume of liquid coming out of the hole per second is
Volume of liquid coming down in the tank per second is
or
If area of opening is much lesser than area of cross section of tank (
A
2
≪
A
1
), then velocity of efflux
which is the same speed that an object would acquire in falling from rest through a distance h under gravity.
The velocity of efflux is the velocity of escaping liquid relative to the container (but not necessarily relative to ground when the container moves).
The formula
tells us if we make two holes at equal vertical distances from top and bottom, both liquids jets will strike the same spot (but not simultaneous).
