Módulos, dormitorios y patios

Let a vessel be filled with a liquid upto a height H and let there be an orifice at a depth h below the free surface of the liquid. The pressure at the free surface of the liquid and also at the orifice is atmospheric, and so there will be no effect of atmospheric pressure on the flow of liquid from the orifice. The liquid on the free surface has no kinetic energy, but only potential energy, while the liquid coming out of the orifice has both the kinetic and

potential energies.

Let P be the atmospheric pressure, r the density of the liq- uid and v the velocity of efflux of the liquid coming out from the orifice. According to Bernoulli’s theorem, the sum of the pressure and the total energy per unit volume of the liquid must be the same at the surface of the liquid and at every point of the orifice. Thus P

+ +

r

+

+

-

r

r

r

This formula was first established in 1644 by Torricelli and is called ‘Torricelli’s theorem’. If a body is dropped freely (u = 0) from a height h, then from the third equation of motion,

v2 _{= 2gh, we have}

v = 2gh

Clearly, the velocity of the liquid falling from a height h is 2gh . Hence the velocity of efflux of a liquid from an orifice is equal to that velocity which the liquid acquires in falling freely from the free surface of the liquid upto the orifice.

After emerging from the orifice the liquid adopts parabolic path. If it takes t second in falling through a vertical distance (H – h), then according to equation s = 1

2 2 at , we have (H

-

-

Since there is no acceleration in the horizontal direction, the horizontal velocity remains constant, the horizontal distance covered by the liquid is

x = horizontal velocity × time

= v × t

= 2gh 2 H h

g

´

-

= 2 h H(

-

This formula shows that whether the orifice in the vessel is at a depth ‘h’ or at a depth (H – h) from the free surface of the liquid, the emerging liquid will fall at the same distance i.e. the range x of the liquid will remain the same.

Now, h (H – h) will be maximum when h = H – h i.e. h = H

2 Hence the maximum range of the liquid is given by

x_max = 2 2 2 H H H H

F

HG

IKJ´FHG

-

IKJ

=

Therefore, when the orifice is exactly in the middle of the wall of the vessel, the stream of the liquid will fall at a maximum distance (equal to the height of the liquid in the vessel).

6.7 VISCOSITY

When a solid body slides over another solid body, a frictional force begins to act between them. The force opposes the relative motion of the bodies. Similarly, when a layer of a liquid slides over another layer of the same liquid, a frictional-force acts between them which opposes the relative motion between the layers. The force is called ‘internal frictional-force’.

Suppose a liquid is flowing in streamlined motion on a fixed horizontal surface AB. The layer of the liquid which is in contact with the surface is at rest, while the velocity of other layers increases with distance from the fixed surface. In the figure,

the lengths of the arrow represent the increasing velocity of the layers. Thus there is a relative motion between adjacent layers of the liquid. Let us consider three parallel layers a, b and c. Their velocities are in the increasing order. The layer a tends to retard the layer b, while b tends to retard c. Thus each layer tends to decrease the velocity of the layer above it. Similarly , each layer tends to

between any two layers of the liquid, internal tangential forces act which try to destroy the relative motion between the layers. These forces are called viscous forces. If the flow of the liquid is to be maintained, an external force must be applied to overcome the dragging viscous forces. In the absence of the external force, the viscous forces would soon bring the liquid to rest. The property of the liquid by virtue of which it opposes the relative motion between its adjacent layers is known as ‘viscosity’.

The property of viscosity is seen in the following examples.

(i) A stirred liquid, when left, comes to rest on account of viscosity. Thicker liquids like honey, coaltar, glycerine, etc have a larger viscosity than thinner ones like water. If we pour coaltar and water on to a table, the coaltar will stop soon while the water will flow upto quite a larger distance.

(ii) If we pour water and honey in separate funnels, water comes out readily from the hole in the funnel while honey takes enough time to do so. This is because honey is much more viscous than water. As honey tends to flow down under gravity, the relative motion between its layers is opposed strongly.

(iii) We can walk fast in air, but not in water.

(iv) The cloud particles fall down very slowly because of the viscosity of air and hence appear floating in the sky. Viscosity comes into play only when there is a relative motion between the layers of the same material. This is why it does not act in solids.