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9
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It is in rad per second
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- At time it has moved through an angle of radians


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For constant angular speed
, rearranging this gives

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The horizontal component of is
Because the rate of rotation is constant, the angle or
is proportional to time
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If the graph is of and time, then it would be and instead
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Topic 9 Page 3

Deriving equations of SHM
SHM is


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This is a differential equation whose solution is:
where

is a constant
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We know that velocity is given by

So for SHM with

For SHM with

Given that

we know that this is a periodic function given by:

The period is the time to complete one full oscillation
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Topic 9 Page 4

Equations of SHM
Displacement
Velocity
Acceleration
It follows that:
- The maximum speed is
- The maximum acceleration is
The use of cosine and sine functions for displacement depends on what the situation is at

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- Use the cosine version when at
the displacement is zero and the particle moves towards positive displacements
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Topic 9 Page 5

is in units rad2 s-2, which is equivalent to

Energy Changes
The total energy is the sum of the potential and kinetic energies at any time:

During oscillations, energy transforms from one form to another
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Topic 9 Page 6

it has kinetic energy only
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When the pendulum bob is displaced from the rest position there is a component of the bob's
weight that restores the bob to its normal rest or equilibrium position
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A condition of a system oscillating simple harmonically is that there is a restoring force that is proportional to the displacement from the equilibrium position
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It can be assumed that the friction between the mass and the base is insignificant
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When a spring is extended by
direction to the extension)
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When the mass is to the left of the equilibrium position the compression is
positive but the restoring force is negative
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Topic 9 Page 8

9
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This causes the wave to spread out past the aperture
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There is no diffraction if the wavelength is much smaller than the opening size

Figure 9
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- Think of the light as wavefronts that are parallel to the slit
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- According to Huygens, every point on the wavefront emits a spherical wavelet of the same velocity and wavelength as the original wave
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- The result of the interference can be seen on a screen placed a large distance away
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10 shows rays from A1 and B1 travelling at an angle to the slit
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- The interference seen at P depends on the path difference, which is B1C1
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- Triangle A1B1C1 is right angled and angle B1A1C1 equals
- The distance A1B1 is equal to
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11 shows that A1, B1, A2, B2 are all parallel
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- Triangles A1B1C1 and A2B2C2 are both right-angled have the same
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For every point on the upper half of the slit there is a corresponding point on the lower half
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Topic 9 Page 10


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We get a minimum at P if:

We get additional minima whenever:

Since the angle

is typically small, we may approximate

The first minimum observed at an angle is given by:

The maxima of the pattern are approximately half-way between minima
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The angular width of the central maximum becomes larger
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When white light is entering a slit there will be different diffraction patterns observed for each wavelength making up the w hite light
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The fringes appear more coloured as we move away from the centre
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are the angles with the straight-through position made by the minima
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3 - Interference

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Young's Double Slit Experiment
Interference takes place when identical waves are emitted from two sources and observe at
the same point in space
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It
diffracted and the turned into plane wavefronts by passing through a lens
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Light diffracts at
each slit and spreads out
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The result is a pattern of bright and dark bands
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Monochromatic light means light of the same wavelength
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Destructive interference occurs when a crest meets a tr ough
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This would make it a constant sine wave
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In double slit, these two effects are superpositioned to produce the double -slit diffraction pattern
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Because of single-slit constructive and destructive interference, the actual double-slit intensity pattern is what you see
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The overall double-slit intensity of the brightest portions of the interference pattern are brighter than those of the single -slit because there are two
sources, rather than one
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The intensity increases with the increase in slits because more light gets through
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At the centre of the principal maximum all the waves from every slit are meeting in phase, making it very bright
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Increasing the number of slits will cause destructive interference close to the centre of the maxima, which reduces the width of the central maximum and other
maxima
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A typical diffraction grating consists of a large number of parallel, equally-spaced lines or grooves,
etched in a glass or plastic substrate through which light passes, or is reflected from
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- is the wavelength
- is the distance between each grate (slit separation)
- is the angle that light incident on the grating produces interference maxima
We cannot use the small angle approximation because the angle is large
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must be converted into
converted to the number of lines per metre
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Topic 9 Page 17

before you can use the equation
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Case 1 - Use data booklet equations
There is a

phase change at the first juncture and no phase change at the second juncture (mark this on the diagram)
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Equation for a maxima (constructive interference):

Equation for a minima (destructive interference):

Topic 9 Page 18

9
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Light diffracts as it goes into the eye of the observer
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Two distant objects that are close together will look like one object if they are far enough away
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This is Rayleigh's Criterion
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If the two soures are too close to each other then their
diffraction patterns will overlap on the retina of the observer, meaning the observer cannot distinguish between the objects
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- is the distance the two objects are separated
- is the distance the two objects are from the observer

For a circular slit things have to be modified somewhat and the angle of diffraction for a circular slit is given by:
For a circular slit, resolution is possible when:

Topic 9 Page 20

where

is the diameter of the slit
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Work out angle

from

(in radians)

2
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The questions may ask you how it is affected in a different medium

Topic 9 Page 21

Diffraction Grating Solution
Two pattern characteristics of intensity patterns for multi-slit interference:
1
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2
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A diffraction grating can be thought of as a collection of narrow slits
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The resolution for a grating will be will be proportional to the number of lines
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It can be said that

, where

is the order at which the lines are observed and

which gives the smallest difference in wavelengths that can be resolved
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So we have:

9
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Moving source and stationary observer
, where
the changed frequency perceived by the observer
the frequency of the stationary source
the speed of sound in air
the speed of the source relative to the observer
The sign is

for a source moving towards the observer and is

for a source moving away from a stationary observer
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Doppler Effect with Light
The equation of the Doppler Effect for light takes a simple form when the speed of the source or the observer is small compared to the speed of light
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If the source of light moves away from the source then the frequency decreases, which is known as a redshift

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