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Consider the sample of material shown below, which is being extended by a force F
.....
The tensile stress

in the material is defined as follows .....

= F/A

-2
The tensile strain

in the material is defined as follows .....

=
 x/ l

( Compressive stress and compressive strain are defined by the same equations as those above. The only difference is that the force arrows are pushing inwards - and
 x is a compression rather than an extension.)
N. B. The advantage of using "stress" and "strain" rather than "force" and "extension" is that a given stress on a given material always produces the same strain in that material. However, the extension produced by a given force is not always the same - it depends on the dimensions of the sample of material.
The strength or breaking stress of a material is the stress at which it breaks. It is a constant for a given material.
A graph of stress against strain for a typical material is shown below .....

Note that it is similar in shape to the force-extension graph in "EAT".
The limit of proportionality is the end of the straight line section. Beyond this point, stress is no longer proportional to strain.
The elastic limit or yield point is where the material makes the transition from elastic to plastic behaviour. Beyond the elastic limit the material no longer returns to its original dimensions when the force (or stress) is removed.
The Young modulus E of the material is defined as follows .....
E = stress/strain =

/

-2
Note that the Young modulus is a constant for a given material . e.g. The Young modulus of steel = 2 x 10 11 Nm -2 . It is a measure of how stiff the material is.
(Recall that: Stiffness k = F/
 x . k is not a constant for a given material. It depends on the dimensions of the sample.)
Since the Young modulus E = stress/strain, it follows that E is also the gradient of the straight part of a stress-strain graph. (It is only valid up to the limit of proportionality.)
The elastic strain energy

E el
stored in a sample of material when it is stretched (or compressed) is the area under the force-extension (or force-compression) graph, as shown below .....

If the material obeys Hooke's Law , the graph is a straight line, as shown below .....
The area then equals the area of a triangle, in which case .....

E el
= F
 x / 2
All particles have wave-like properties . i.e. They can be diffracted, interfere with each other, etc. In practice, the wave effects are only noticeable with very small particles (such as electrons). The phenomenon is often referred to as "waveparticle duality" .

N.B. These waves are not electromagnetic waves. It is simply that the behaviour of the electrons etc can be explained by using a wave model , and using ideas such as superposition etc.
A particle's wavelength can be calculated using the formula

= h / (the particle's momentum) = h/mv ..... where h = Planck's constant. This wavelength is sometimes called its de Broglie wavelength .
Electrons accelerated in an "electron gun" towards a positive anode at about 1000 V, have wavelengths of the order of 4 x 10 -11 m. The atoms in a solid are ~ 2 x 10 -10 m apart. This means that the electrons' wavelength is of the right order of magnitude for diffraction to occur when they are fired at a solid. (It's similar to the wavelength of Xrays.) In other words, the wavelength is smaller than the "aperture" (the gap between the atoms) - but not too small.
The diagram below shows what happens .....
If the solid is crystalline, as shown above, the diffraction pattern consists of "dots".
The "dots" are places where many electrons arrive at the screen. In terms of the wave model, they are places where the wave amplitude is high.
A pattern of dots indicates an ordered structure - such as a crystalline material.
If the material is disordered at the atomic level (e.g. a liquid, or an amorphous material - like glass), the pattern is one of concentric rings, such as that shown below
.....

A pattern of concentric rings indicates a disordered structure.
The difference at a molecular level between an unstretched and a stretched polymer
(such as rubber, plastics, etc) is shown below .....
Thus the unstretched polymer is a disordered structure, and would therefore have a diffraction pattern of concentric rings. The stretched polymer, however, is an ordered structure - its diffraction pattern would consist of "dots".
In summary, electron diffraction can be used to determine the structure of materials.
You now have to know about diverging lenses - as well as converging lenses.
The principal focus F and focal length f of a diverging lens are defined as follows
.....

The power P of a diverging lens (of focal length f) is given by the same formula as before .....
P = 1/f
N.B. Since f is negative, P will be negative .
The same lens formula also applies .....
1/v + 1/u = 1/f
N.B. You will find that v always turns out to be negative for a diverging lens
(assuming the object is real - i.e. u is positive). Negative v indicates that the image is virtual . i.e. The image cannot be formed on a screen (but can be seen by the eye).
A typical ray diagram for a diverging lens would be as follows .....
The image is "virtual" in the sense that rays of light are diverging away from it - so

there appears to be something there as far as the eye/brain is concerned.
In short-sightedness (myopia) the image of a distant object is formed in front of the retina. A diverging lens is used is used to throw the image further back (i.e. to diverge the rays).
In long-sightedness (hyperopia) the image of a nearby object is formed behind the retina. A converging lens is used is used to bring the image further forward (i.e. to converge the rays).
The correction of eye defects is illustrated below .....
When waves of any kind hit an interface (or boundary) between 2 different media some of the incident wave intensity I i
is reflected off the boundary, and some is transmitted through the boundary, as shown below .....

(Intensity is defined as "power per unit area" ..... Wm -2 )
Ultrasound is very high frequency sound (too high to hear), which is used in medicine to get a picture of what is inside the body. (It has the same speed in a given medium as "ordinary" sound.) The fraction of ultrasound reflected (or transmitted) depends on how different the acoustic impedances Z of the 2 media are.
Z is given by the formula .....
Z =
 v
..... where

is the density of the medium, and v is the velocity of sound in the medium.
If the acoustic impedance of medium 2 is very different from that of medium 1, only a very small fraction of the ultrasound intensity will be transmitted - and a large fraction will be reflected. (In other words, I t
/ I i
will be small, and I r
/ I i
will be large.)
If the 2 acoustic impedances are similar, the reverse will be true (I t
/ I i
will be large, and I r
/ I i
will be small.)
Air and soft tissue have very different acoustic impedances, so we don't want an air gap between the ultrasound probe and the surface of the patient's skin. This would cause the ultrasound to be reflected back into the probe. For this reason, a coupling gel is used, which has a similar acoustic impedance to soft tissue (and to the probe).
See below .....

On the other hand, we do want the ultrasound to be reflected off structures inside the body that are being examined. This will happen provided their acoustic impedances are very different from that of the surface layers of soft tissue. e.g. Airfilled organs and bone are good reflectors.
The ultrasound probe (or transducer ) often acts as both transmitter and receiver.
Consider the situation below .....
Some of the ultrasound may be reflected off the skin surface, some off the top of the organ, and some off the bottom of the organ. The received signal would be as shown below .....

t
B
= the time taken for the ultrasound to travel from the surface of the skin to the top of the organ and back again . t
C
= the time taken for the ultrasound to travel from the surface of the skin to the bottom of the organ and back again .
Thus the depth of the top of the organ would be calculated as follows ..... depth = (speed of ultrasound in soft tissue) x 1 /
2
t
B
Such information can be processed using a computer to produce an image of the internal structures within the body. To get a moving image you need to send the ultrasound in short pulses and refresh the image many times per second.
When you move towards or away from a source of waves (light, sound, etc.) - or the wave source moves relative to you - there is a change in the wavelength and the frequency of the waves. This phenomenon is called the Doppler effect .
The diagram below illustrates the effect .....

In summary ....
When the source moves towards the observer - or the observer moves towards the source .....

decreases and f increases
When the source moves away from the observer - or the observer moves away from the source .....

increases and f decreases
(The diagram above explains the Doppler effect for a moving source. If the observer is moving, the explanation is as follows: when the observer is moving towards the source, more wave-crests are encountered per second - so the frequency appears higher and wavelength appears shorter. Vice versa if the observer is moving away from the source.)
Note also that the velocity of the waves is not affected by the Doppler effect. In the formula ..... v = f


..... v is a constant .
So, if

increases, f must decrease ..... and vice versa.
There is a formula for the wavelength change

and the frequency change
 f .....

=
 f / f = u / v
..... where

and f are the original wavelength and frequency, u is the relative velocity, and v is the velocity of the waves.
If the observer transmits waves and then receives them after they have been reflected from a moving object, as shown below .....
..... then you get a "double Doppler effect" . i.e.

=
 f / f = 2u / v
If the "gun" is at angle

to the direction of motion of the reflector, as shown below
.....

..... then the formula becomes .....

=
 f / f = 2ucos

/ v