## Resolvance of a Diffraction Grating

Illuminating a diffraction grating with monochromatic light from a He/Ne laser shows a typical pattern, out in the photograph to m=3 on both sides. The spots are equally spaced and we notice that the m=2 spot is hidden under the first single slit diffraction minimum – a “missing order”. The geometry is identical to that for a double slit, d being the distance between the centre of one slit and the next. For a bright maximum:  Unlike two-slit interference, only at very particular angles do the contributions from each slit add constructively. Everywhere else, the contribution from one slit has a partner somewhere else down the grating which cancels its contribution out, hence the very bright spots and a lot of empty space.

You are strongly encouraged to go to the Wolfram Demonstrations Project, download the CDF player and experiment with this demonstration. 1, 2 or many slits -the choice is yours. With 15 slits the pattern is almost indistinguishable from a diffraction grating – screenshot below – the single slit diffraction envelope is clearly shown. Light intensity (y-axis) is proportional to amplitude squared. A flame test for sodium displays a very bright yellow emission. This emission is due to the sodium D-lines – two lines very close together. The diagram shows the absorption spectrum of the Sun by Fraunhöfer who labelled the lines. The sodium doublet is seen at wavelengths of about 589.0 nm and 589.6nm.

How could these be resolved using a diffraction grating? We recall that a diffraction grating gives sharp, clear orders.

More accurately, the D lines have wavelength1 = 589.592nm and wavelength2 = 588.995nm

We can find the resolvance or the resolving power required for the doublet to be resolved. For N lines of the diffraction grating, we can write (without derivation) for the mth order: So, in this case, for a required resolvance of about 1000, viewing the second order would need N=500 grating lines to be illuminated – even the coarsest of gratings manages this easily – a grating with 1800 lines per mm is quite common, if rather expensive. The larger N the better the resolution. If third, fourth or greater orders are visible, a coarser hence cheaper grating will do.

## Newton’s Laws of Motion Fun fact. Newton laughed only once in his life, when somebody asked him what was the point of studying Euclid.

FIRST: ” A body continues in a state of rest or motion at constant speed in a straight line unless acted upon by an unbalanced external force.” SECOND: “The applied force is equal to the rate of change of momentum of the body.” . A rather more modern interpretation is here. If cliff-diving appeals to you, watch this video. As long as you don’t scare easily… the conveyor belt problem – if we are to keep a conveyor belt moving at a steady speed – for example in a coal mine where mass is being added to it all the time, we require a force to be applied to the conveyor belt.

THIRD: “For every action, there is an equal and opposite reaction”

Think about a jet propulsion system. Thrust is a mechanical force which is generated through the reaction of accelerating a mass of gas, as explained by Newton 3. A gas or working fluid is accelerated to the rear and the engine and aircraft are accelerated in the opposite direction.

The force on the working fluid is equal and opposite to the force on the engine and aircraft. Look here for a very easy walkthrough of all of Newton’s Laws. This is a particularly good treatment so you can work through the videos yourselves.

## Doppler Effect

Imagine a Formula 1 car approaching the stands at 60m/s. The frequency of sound made by the engine as heard by a stationary observer in the stands is higher than the actual frequency as heard by the driver. The sound is squashed up – or better, the apparent wavelength is decreased and the apparent frequency increased.

As the car recedes away from the stands, exactly the reverse happens. the observer waves goodbye to the red line. Think of EEEEYOWWWW as the car approaches then recedes.

For a stationary observer and a moving source, we can write: These will mostly do – but IB requires us to use these as well:- A quick calculation shows how the first equation works. Let the car be moving towards us in the stands at a speed us of 60m/s and emitting a frequency f of 800Hz.

Speed of sound in air is 340m/s. We can find the frequency f’ as heard by the stationary observer. Common sense tells us whether we add or subtract the velocities – in this case, we subtract and hear a higher frequency as it approaches us. (the EEEE bit) As it recedes, we add, thus: (the YOWWW bit) Police speed detectors bounce microwave radiation (about 10GHz) off a moving vehicle and detect the reflected waves. Because the car is moving towards the police observer, these waves are shifted in frequency by the Doppler effect and the difference in frequency between the transmitted and reflected waves provides a measure of the vehicle’s speed. Of course it works just as well for recession speeds as well. Two Doppler shifts because of the reflection from a moving target. c is of course the speed of light By observing distant galaxies, Edwin Hubble concluded that distance and recession speed were proportional – so galaxies further away are receding faster than closer galaxies. We know this because the atomic fingerprint or spectrum of atomic hydrogen or helium is shifted to the red (long wavelength) end of the visible spectrum. The degree of redshift can be used to find out how far away a galaxy is. This absorption spectrum shot (idealised) shows what the spectrum of atomic hydrogen might look like from several distant objects like galaxies. The further away, the greater the redshift. Redshifts of up to 0.95c have been observed – the light having taken almost the lifetime of the Universe to reach us. Finally, a medical use. Doppler blood flow is a technique whereby ultrasound waves (f about 800Hz) emitted from a piezoelectric transducer (transmitter/receiver) are reflected off red blood cells in an artery or vein as they are moving towards the stationary detector. The more occluded or blocked the artery is (think about a fluid in a pipe) the faster the cells are moving. It can also be used to find blood clots in deep veins – DVT – deep vein thrombosis – can be fatal. The detector and the moving cells are at an angle hence the cosine term and, like the police car, the factor 2 accounts for the reflection from a moving source. ## IB: Engineering Science Option B: Torque, Angular Momentum and Moment of Inertia (amended)

THIS POST  WAS ORIGINALLY WRITTEN FOR MY OWN IB CLASS. THERE ARE HANDOUTS AND PROBLEMS HERE THAT WE DID IN CLASS BUT NEWCOMERS SHOULD FIND THEM HELPFUL. GO AHEAD AND TRY.

It’s useful to bear in mind that if you can do SUVAT problems, you should have no trouble with their circular equivalents.

Four handouts in total to download; please make sure you work through them carefully. Any difficulty, get in touch.

The main arguments here are the idea of rotational motion and torque as force x distance from pivot x sin(angle between them)

Moments of Inertia need not be calculated for this course – if necessary, you’ll be given them. However, here’s a little problem to think about. You have 2 balls of identical diameter and weight. One is solid, one is hollow. You can’t tell which is which just by knocking on them. Devise a simple way of finding out which one is the solid one (hint: think about the balls rolling down an inclined plane from the same height. Now, compare the moments of inertia of the two balls. The rest is conservation of energy so quite easy.) If you can’t write out the solution, message me for help.

This little animation is quite fun to glance at – the angular displacement is, however, in degrees, not radians, so be careful.  Here’s a screenshot, showing displacement – time graphs for the ant and the ladybug – the constant time period implies constant angular velocity. Notice too that the ladybug leads the ant by 90 degrees Notice, angular velocity is constant, but the linear speed of the ant and the ladybug are not the same. The ladybug, being closer to the axis of rotation has a smaller linear speed, because: (Notice the vertical displacements of the bugs execute SHM – AHLs will study this later)

1. A review of circular-motion-1
2. Key Ideas, torque and couple
4. From the Specimen Paper specimen-question-for-option-b

Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis. Just as inertial mass is the ratio of linear momentum to speed – its resistance to acceleration, in other words: Angular momentum in a closed system is, just like its linear counterpart, conserved. The ice skater rotates faster when the arms drop to the sides because moment of inertia is reduced and thus angular velocity increases.

Look at the Wolfram demonstration You will need to download the Wolfram CDF player in order to run the demo.

We might also notice that, for a body starting to rotate from rest: Practically, we find I by imagining a flat sheet of any shape like this having an infinite number of mass elements m at their respective distances r from the pivot, each contributing  torque about the axis of rotation.

We have to add all the torques up, normally requiring integration. But, to keep it simple, we can write: which is, as the handout shows, the basis for finding I for lots of other shapes and axes of rotation. Remember, you’ll be given I for a particular shape as required. Bear in mind that when we do problems, the total energy of the system is the sum of the rotational and linear parts – important when we think about an object rolling (instead of sliding) down a hill, for example. Take a look at this solid-cylinder-rolling-down-an-inclined-plane. which runs through a few basic ideas plus some possible lab work.

Finally, for now, a use for all that stored energy. The great flywheel on Richard Trevithick’s 1802 locomotive, used to level out the power supplied by a single cylinder. Rotational inertia kept the wheel turning.

## Charge Coupled Devices – Camera Basics

A CCD contains a capacitative array which individually photoelectrically converts incoming light photons into electrons, hence a voltage across each capacitor, whose magnitude is dependent on how many electrons were released by the incoming illumination. One pixel is one of these capacitors. These voltages are processed as an array, to yield a digital image. Clearly, the solid state capacitors are sensitive to the spectrum of visible frequencies, and electrons are emitted irrespective of the intensity of incident light. If a 9Mpx camera has a square CCD, measuring 3cm x3cm, each side has length 3000px, so each pixel is a square of side 0.01mm. If  two objects are to be resolved by the array, their images must be formed on the camera at least two pixels apart, achieved using the optics of the camera. Quantum Efficiency of a pixel is the ratio of number of incident photons of a particular frequency to the number of emitted electrons. Higher efficiency means that for a particular intensity of illumination, more electrons are emitted so the image is clearer and brighter. Low light images are clearer and signal to noise ratio is increased.

Ask yourself a question. If this camera has a magnification of 0.01 and is to capture an image of two sticks 5mm apart, will they be resolved, or not?

Answer – the image on the CCD is 0.05mm – much bigger than 2 pixels wide. So, the sticks will be resolved.

## How big is the earth? It’s amazing how coincidence and a little critical thinking goes a long way. Homer the poet believed the Earth to be a flat disk, so if you walk far enough, you’ll fall off the end. Pythagoras believed it was a round ball, but how big was it? Eratosthenes, a Greek scholar living in Egypt 2,300 years ago had heard that on the longest day of summer the midday sun shone right to the bottom of a well in the town of Syene which was near what is now the Aswan Dam. At the same time, he found out that the sun was not directly overhead to the north in Alexandria, the shadow cast there wasn’t vertical instead, it cast a shadow with the vertical equal to 1/50th of a circle (7° 12′). He ‘knew’ that: 1) on the day of the summer solstice, the midday sun was directly over the Tropic of Cancer; 2) Syene was on this tropic; 3) Alexandria and Syene lay on a direct north-south line and 4) the sun was a relatively long way away so the rays of the sun didn’t spread out like the beam of a torch but were parallel instead. According to legend, he had someone walk from Alexandria to Syene to measure the distance – no afternoon stroll, for sure – it turned out to be 5000 stadia or, at 185 m per stadion, about 925 km, but I can’t imagine anybody doing a repeat measurement for accuracy since it would have taken over a month to make the trip. How long was a stadion? Various theories exist, but a good approximation might be that one Roman mile is 8 stadia, or ‘stades’ or 5000 ‘Roman feet’.  From these observations, he thus concluded  that, since the angular deviation of the sun from the vertical at Alexandria was also the angle of the subtended arc, the linear distance between Alexandria and Syene was 1/50 of the circumference of the Earth which thus must be 50×5000 = 250,000 stadia or just over 40,000km. The diagram isn’t precise – the Tropic isn’t this far north, but, you get the idea. The Roman mile is shorter than its present day counterpart, in case anybody noticed. At the Equator, the Earth is 24, 902 modern miles round. More or less.

Just for completeness, here’s a fun little exercise on finding the circumference using data from the space station.

Now we know how big it is, we can use Newton’s Law of Gravitation to find its mass. Imagine a 1kg mass anywhere on the circumference r, found because we now know its circumference. G and g are both known so its mass drops out nicely. Having found its mass we can use density =mass/volume to find the average density of the material of the Earth. 