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Viscosity
Fluids with high viscosity (e.g. honey, treacle, etc) flow slowly through pipes, out of
containers, etc. When a body moves through a very viscous fluid there is a large
viscous drag force. The reverse is true for fluids with low viscosity (e.g. water, etc).
The coefficient of viscosity  of a fluid tells us how viscous it is.  can be related to
the viscous drag F on a sphere of radius r travelling at constant velocity v through
the fluid, as shown below:
The formula that relates the variables is called Stokes' Law:
F = 6rv
The units of  are Nsm-2.
e.g.
viscosity of water ~ 1.0 x 10-3 Nsm-2
viscosity of air ~ 1.8 x 10-5 Nsm-2
The viscosities of most fluids are very sensitive to temperature:
 goes down as temperature goes up
Terminal Velocity
Consider a ball-bearing falling through a liquid. It has 3 forces acting on it, as shown
below:
The weight W is a constant force. It is given by:
W = mg = density of steel x volume of sphere x g = 4/3r3steelg
density of steel
where steel =
The upthrust U is also a constant force. According to Archimedes' Principle:
4/
U = weight of fluid displaced = density of fluid x volume of sphere x g =
3
where fluid = density of fluid
fluidg
3r
The viscous drag F, however, increases as the velocity of the sphere increases.
According to Stokes' Law:
F = 6rv
Assume the ball starts from rest at the surface of the fluid - so initially the viscous
drag F = 0. Thus, provided W > U, the ball will accelerate downwards to begin with since there is a net downwards force. However, as the ball accelerates, v increases so the viscous drag F increases ..... until eventually .....
U + F becomes equal to W
At this point there is no net force on the ball - so it no longer accelerates. It has
reached terminal velocity.
Substituting all the above expressions into ......
U+F=W
(which is true at terminal velocity)
we get .....
4/
3
3r fluidg
+ 6rv = 4/3r3steelg
An experiment can be done to measure the terminal velocity v, and the above
equation can then be re-arranged to find .
Streamlined and Turbulent Flow
In streamlined or laminar flow the fluid does not make any abrupt changes in
direction or speed. It may be thought of as layered flow.
In turbulent flow there is chaos, mixing, eddies, etc.
Both cases are illustrated below:
Turbulence often sets in at higher flow rates or if the fluid has to flow around a shape
that is not "aerodynamic" (e.g. a block etc.). It is normally undesirable because it
heats the fluid and is therefore inefficient.
Properties of Materials
Materials deform (i.e. extend or compress) when forces are applied to them.
In elastic deformation the material returns to its original dimensions when the the
force is removed.
In plastic deformation the material remains stretched or compressed when the
force is removed.
Most materials are elastic for small loads (forces) - but behave plastically if the force
is increased beyond a certain point. This is illustrated by the graph below, which is
for a typical material (e.g. copper wire):
If the load is removed in the elastic region (0 - A), the material returns to 0 (zero
extension/compression).
If the load is removed in the plastic region (between A and B), the material returns to
C. i.e. It is permanently deformed.
Ductile materials can be drawn out into long thin shapes (or wires) - e.g. chewing
gum.
Malleable materials can be deformed (hammered) into a flat shape - e.g. fudge.
Ductility and malleability are both examples of plastic behaviour.
Brittle materials crack and shatter easily. They are usually stiff, and break before
becoming plastic - e.g. glass.
Tough materials are not brittle. They can withstand dynamic loads (shocks). They
also deform plastically (and absorb a lot of energy) before breaking, Tough materials
in the engineering sense are usually stiff as well - e.g. Kevlar.
Stiff materials (e.g. steel) need a large force to produce a small deformation. i.e.
Their stiffness k is large (see HFS notes).
Hard materials are resistant to scratching and indentation - e.g. diamond.
Refractometry
Refractometry can be used to determine the concentrations of sugar solutions by
measuring their refractive index.
The diagram below shows a method for measuring the refractive index of a small
amount of a liquid - such as a sugar solution:
With the eye at A the mark on the glass (X) is not visible, because the light reaching
the eye has been totally internally reflected at X - and therefore comes from a point
D (not from the mark itself).
With the eye at B the mark is just visible because the ray from X to the eye is at the
critical angle (C) to the normal at X.
With the eye lower than B the mark is easily visible because the light has come from
the far side of the glass-liquid boundary.
The position B is found and marked with pins. The angles C, r and i are measured.
The critical angle C is for the liquid-glass interface.
 lg = sin 90 / sin C = 1 / sin C
index
Also:
..... where lg is the liquid to glass refractive
ag
= sin i / sin r
..... where ag is the air to glass refractive index
and .....
ag
= al x lg
We require al
al
..... the refractive index of the liquid
= ag / lg
 al = (sin i / sin r) x sin C = (sin i / sin r) x cos r = sin i / tan r
The refractive index of known concentrations of sugar solution can be measured and
a graph plotted, as shown below:
The sugar concentration of an unknown solution can then be found by measuring its
refractive index and using the above graph.
Polarisation
Light is part of the electromagnetic spectrum. That is to say, it is a transverse
wave consisting of sinusoidally varying electric and magnetic fields at right angles to
each other. It is "normally" unpolarised. In other words, the oscillations (of the
electric and magnetic fields) are in every plane at right angles to the direction of
travel - as shown below:
In plane polarised light, the oscillations are confined to one plane - as shown below:
(Strictly speaking, the above wave just represents the variation of the electric field so there should be another wave at right angles to it representing the variation of the
magnetic field. However, it is usual to omit the magnetic wave for the sake of clarity.)
Unpolarised light can be polarised by passing it through a polarising filter (such as
"Polaroid"). Light reflected off glass, water and many other surfaces is also partially
plane polarised.
Polarimetry
Certain materials (sugar solution being an example) rotate the plane of polarisation
of light. In the case of sugar solution, the higher the concentration the greater is the
angle of rotation. This fact can be used to measure the strength of a sugar solution using a polarimeter, as shown below:
The cylinder is filled with pure water to begin with, and the top polarising filter is
rotated until the plane of light that it allows through is at right angles to that allowed
through by the bottom filter - i.e. until it blocks the polarised light produced by the
bottom filter. The cylinder is then filled to a certain depth with the sugar solution.
Since the sugar solution rotates the plane of polarisation, the light is no longer
completely blocked, and it is necessary to rotate the top filter in order the block the
light again. The angle through which the top filter has to be rotated is the angle
through which the plane of polarisation has been rotated. A graph of "angle of
rotation" against "sugar concentration" can thus be plotted, as shown below:
(N.B. You have to use the same depth of sugar solution each time because the
angle of rotation also depends on the depth.)
The rotation angle of an unkown solution can be measured, and the sugar
concentration read off the graph.