Centre of Mass (or Gravity)

Imagine you could squash a pool ball up to the size of a pinhead. An arrow drawn vertically downwards from that point shows the centre of gravity, acting at the centre of mass.

Where might the centre of gravity of a pool cue be? Obviously at a point nearer the handle end where the wood’s diameter is greater.

Finding C of M is easy. Cut out a funny shape from a cereal packet. Make a tiny hole in one corner with a pin, making sure the piece of card can move freely and hang a piece of cotton with a weight on the end on it. Draw a line down the vertical string. Do exactly the same somewhere else on the card. Where the lines cross is the centre of mass. The force of this mass acting downwards acts along the line of the centre of gravity.

Some objects have a centre of mass outside of the material of the object, like  a wooden lab stool. All the mass is in the seat, so the centre of mass is in thin air somewhere underneath the seat.

Where’s the centre of mass of a Bunsen burner? Probably in the metal bit at the base. It’s in stable equlilbrium so if it’s tipped it’ll fall back on to its base again.

Remember the OSMIUM ROCKER from the density post. Think about what would happen. You might need your density table handout. Look back at it and suggest what might happen if you sat on the rocker

Can you balance a potato on a knife edge? Sure you can. Two forks lower the centre of gravity to below the knife edge so that if it’s tilted, it’ll tend to return to the equilibrium position on the knife edge. Try it at home…ASK FIRST.

You might think about this as the ‘tightrope walker principle’. The only reason tightrope walkers don’t fall off is that the massive pole they carry lowers their centre of mass so it’s below the rope. in fact, they CAN’T fall off. 

In 1859, a man called Blondin first walked across Niagara Falls on a tightrope. He did it lots of times, once carrying a man on his back! Notice the pole!

 

 

 

 

 

 

London buses can tilt an awfully long way before toppling over also. See a bus doing the tilt test here. Why doesn’t it topple over? Because the centre of mass is low, almost between the wheels, so as long as the centre of gravity acts inside the pivoting wheel, it won’t topple over.

Now, something for you to do. Tails on birds can lower their centre of mass so they can perch on wires. Cut out a parrot like this one from a piece of card about 25-30 cm high. Weight the tail with a heavy clip – it helps if you curl the tail down underneath the bird a bit. See if your parrot can stand on a wire on his own… Colour it in and bring them all to school and we’ll display them.

Ladders and Tramlines: Electrons as waves and particles

What’s an eV?

eV stands for electron volt – that is, the amount of energy needed to move one electron through a potential difference of one volt. If that doesn’t mean much to you, don’t worry about it; the point is that it’s a small unit of energy, convenient for the scale of energy levels in an atom. To give you an idea of just how small an eV is…a 100 watt light bulb is putting out 100J of energy per second. That’s about 625,000,000,000,000,000,000eV per second!You may also have run across KeV (kilo-eV; that’s one thousand eV) and Mev (mega-eV, which is one million eV) in the med phys course.

Put another way, if we shine light on potassium, for example, photoelectric emission only happens if the energy of the incident light-bullets (or PHOTONS) is greater than 2.1eV – the work function energy of K Anything less, the electrons in the metal just get agitated.

Converting from eV to J:

2.1 x 1.6 x 10-19J = 3.36 x 10-19J is the minimum energy a light-bullet has to have to hook an electron out of the potassium surface. If the photons have more energy than this, the extra is picked up as the KE of emitted electrons.

Given h = 6.63 x 10-34 Js and E = hf = hc/λ, this corresponds to a wavelength of 592nm – a yellow-green colour.

Electrons in atoms

We know that electrons don’t just wander about – instead they exist in well-defined energy levels, much like the rungs of a ladder define how much gravitational potential energy it takes to climb them. But, the rungs aren’t evenly spaced. The ones near the nucleus are bigger jumps than the ones further away. For hydrogen the biggest jump an electron needs to get out is 13.6eV  – the first ionisation energy of hydrogen. If given 13.5eV, the electron just sits there, vibrating mutinously. To get it out it’ll have to get EXACTLY 13.6eV of energy.

These rungs of energy are a little bit like tramlines, the electrons almost run on rails. Why do electrons exist in these ‘tramlines in the sky’?

Because the wavelength of the electron has to fit inside the orbital like a curved guitar string, setting up a standing wave around the orbital. Any old orbital won’t do – only those allowing a whole number of wavelengths to fit is allowed. If you look carefully at the picture, you can see the circular standing waves on the loop of wire which is being driven by a small oscillator box at the bottom This is exactly how the electrons behave when orbiting a nucleus.

Here’s the ‘energy ladder’ for hydrogen. Remember, no intermediate steps are allowed. Each step corresponds to the emission of a photon having the same energy as the ‘jump’ between the rungs. Those falling to n=1 are in the UV, (Lyman series) those falling to n=2 are the Balmer series, shorter, visible wavelengths. n=1 is sometimes called the GROUND STATE and in books is sometimes given a – sign denoting that energy is required to be given to the electron (think of it stuck down a well and has to climb out).

Lyman is misspelled...
Lyman is misspelled...

Polarisation, Brewster’s Law and Malus’ Law

If a wave is plane polarised, all the oscillation directions of the electric or magnetic field vector  (it’s important in an exam to actually say what’s oscillating)  except one are absorbed. This can only happen with transverse oscillations since these are oscillations at right angles to the direction of travel and there are lots of ways for (example) an EM wave can oscillate perpendicular to the travel direction; in a longitudinal wave, there’s only one.

This flash animation shows a polarised EM wave, the electric field is restricted to one direction.

This shows what happens when a polariser is placed in the path of an unpolarised wave. The direction of oscillation is restricted to one. Another crossed polariser reduces the intensity to zero.

Using 3cm microwaves, we see something like this. The microwaves produced from the klystron are already plane polarised – vertical electric oscillations only. We can direct them towards a metal grid, arranged vertically, as shown. We might imagine that the polarised waves would pass through the grid, much like a wave on a string would pass through the gaps. Quite the reverse happens- the energy is absorbed by the free electrons in the metal of the grid, setting up a stationary wave in the vertical bars, thus absorbing the energy. When the bars are horizontal, the microwave energy passes through unimpeded.

Applied stress rotates the plane of polarised light. This image shows what happens when a plastic hook is loaded and viewed through crossed polaroids. This phenomenon, known as photoelasticity was discovered by David Brewster (see later). In summary, the effect is due to the fact that the magnitude of the refractive indices at each point in the material is directly related to stresses experienced by the material at that point.

Light reflected from surfaces like a flat road, smooth water or glass is generally horizontally polarized. This horizontally polarized light is blocked by the vertically oriented polarizers in Polaroid sunglass lenses, substantially reducing glare.

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Same house, right hand photo through a polarizer.

At a particular angle, however:

Reminding ourselves that weak reflections accompany refraction, when light is refracted at an interface and the reflected and refracted components are at 90 degrees, the surface acts like a polarizer – reflected component is perfectly polarized one way, refracted is partially polarized the other. This happens at Brewster’s angle.

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For a glass/air interface, and using Snell’s Law, Brewster’s Angle is about 56 degrees.

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Malus’ Law  says that when a perfect polarizer is placed in a polarized beam of light, the intensity, I, of the light that passes through is given by

 Screen Shot 9.png

where I0 is the initial intensity, and θi is the angle between the light’s initial polarization direction and the axis of the polarizer.

A beam of unpolarized light can be thought of as containing a uniform mixture of linear polarizations at all possible angles. Since the average value o \cos ^{2}\theta is 1/2, the transmission coefficient becomes

{\frac {I}{I_{0}}}={\frac {1}{2}}.

Put another way, if a polarizing filter is placed at some angle theta to a vertically oriented plane of polarized light, its amplitude E will be reduced by a factor cos theta

Screen Shot 10.pngScreen Shot 11.png

and since intensity is amplitude squared, then, as above, Screen Shot 9.pngThis is known as Malus’ Law.

It can be determined experimentally quite easily. Taping a piece of Polaroid (oriented so as to pass only the vertical component) over a beam of white light from an overhead projector will polarize the beam. Orient a second Polaroid at the same vertical angle and rotate it, using a calibrated LDR circuit to measure the intensity (seen here as a current), taking measurements at suitable angles, all the way around to 360 degrees.

The graph will look something like this, peaking at 0, 180 and 360 (cos squared=1) and zero at 90 and 270, cos squared =0.

 

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The Inverse Square Law

 

 

Point source?
Point source?

This works for all waves but it’s convenient to think about EM waves. Imagine a point source of EM wave energy, such as a candle, a light bulb or even the Sun, radiating energy in all directions. We know that  a sphere has surface area = 4πr². If we move twice as far away, the energy is then smeared out over an area FOUR times the size, reducing the intensity by a factor of FOUR. Thus, intensity is inversely proportional to the square of the radius.

 

 

Here’s a problem. Imagine a 40W light bulb, which we can approximate to a point source. How far away would we have to be for the intensity (in watts per square metre) to drop to a) 5mW/m² b) 0.05mW/m²?

Answers:

a) 40W/0.005W = area of the sphere = 8000m² = 4πr², hence r=25.23m

b) 40W/0.00005W = area of sphere = 800,000m², thus r = 252.3m

If I could measure the intensity at a distance of 100km, what would it be?

Answer:

r = 100,000m,  I = 40/4π.(100,000)² = 0.32nW/m². Is this measurable?

Waves within Boundaries : Strings

A guitar string is fixed at both ends. If we pluck a guitar string in the middle, near the twelfth fret, the string oscillates with SHM. The tension and mass per unit length defines how fast the string moves in both directions down the string to either end, where it reflects with a phase change of π radians, superposing with the wave coming the other way. This sets up a resonant condition in the string, each part of the string oscillating in phase with variable amplitude down its length, maximum amplitude being in the middle. The string length defines the fundamental frequency ( v = fλ) and is half a wavelength . Plucking the first harmonic on the twelfth fret sets up a standing wave an octave higher, with a NODE or position of no disturbance in the middle and antinodes on either side.

The fundamental, and first to fourth harmonics are shown.

 

Nodes (x=0) are at each end
Nodes (x=0) are at each end

Huygens and Single Slit Diffraction

Huygens’ Constructions are geometric diagrams which are a fast way of getting to grips with several progressive wave properties like reflection, refraction, diffraction and interference, particularly with EM waves, but modelled in a ripple tank. Here’s an overview.

A plane wavefront consists of a large number of secondary circular wavelets whose speed is constant, if the medium is isotropic (uniform density, like glass or water). These superpose algebraically to form a new wavefront elsewhere in time and space.

Let’s look at diffraction at a single slit. When we see diffraction effects in a ripple tank, we see something like this…the wave smears out horizontally.

The most circular diffraction occurs when the gap size is of the order of 1 wavelength.

If we try the same trick with monochromatic laser light, however, we don’t just see the wave energy ‘smearing out’, as we can see here. Instead, we see a series of light and dark bands.  This applet shows the simplest single slit diffraction very well, but you need the Wolfram CDF player to run it. You can change the colour (wavelength in air) of the light and also change the width of the slit. If we use red light, the pattern is broader than with blue light. Of course, we can’t physically do this, since we usually only have a laser with one colour available. Wider slits allow more light in but produce a narrower set of light and dark bands. Think about what it would look like if we were able to use light of any color, and be able to vary the slit width. Notice the central bright maximum is twice as wide as the others.

Screen Shot 6
Single slit diffraction from a He/Ne laser wavelength 589nm

Now, how does it work? Imagine dividing the slit into two and the slit to consist of lots of secondary wavelets, as in the diagram. The diffracted rays can be thought of as being parallel because the screen is a very, very long way away.

Screen Shot 4

Now think about the first dark fringe. The light from a wavelet near the top end of the slit has a partner exactly halfway down the slit with a path difference of exactly half a wavelength.

The next wavelet down is similarly partnered, and so on all the way down the slit. This means that  every pair of wavelets is superposing destructively, or cancelling out, causing the first dark fringe.

Screen Shot 3.png

in some syllabuses written as:

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Similarly, the light bands are formed because each wavelet is partnered with another with a path difference of  a whole wavelength.so, this time,

 

 

Screen Shot 5

or:

Screen Shot 1

 

We notice that the bright bands are much less bright, which this little construction explains. Imagine this time that the slit is divided into three. At a particular angle, two-thirds of the slit has a wavelet partner half a wavelength out of phase, leaving just one third of unpartnered contributions to form the first or subsequent maxima, so it is much less bright than the central bright maximum.

Screen Shot 1

Additional thoughts about Interference and Diffraction

Interference happens when wave displacements (amplitude = max displacement) add together algebraically in space and time. This is a fancy way of saying that if two waves/pulses/ripples meet, their amplitudes add up at a particular place and a particular moment. When two crests meet, we get – what a surprise, a BIG crest – interference is CONSTRUCTIVE. When a crest meets a trough, we get zero disturbance- DESTRUCTIVE interference. Easiest to see with circular water waves because the wavelengths are of the order of centimetres.

Diffraction is the spreading of a wave round an edge. There’s only a change in the wave direction at the edge of the gap(s). If we imagine (à la Huygens) that a plane wave consists of a  infinite number of closely packed circular ripples, at the edge of the gap, there’s nowhere else for the wave energy to go except in a circle. NB: the most circular pattern is observed when the wavelength is the same size as the gap. This applies whatever wave type we think about. It explains sound diffraction around doorways,  also radio wave diffraction in the Welsh hills, so even the Welsh get to watch the telly….Shorter wavelengths won’t diffract as well around large obstacles, so if the gap size is large compared to the wavelength, we don’t see much diffraction.Screen Shot.png

The thing to now get hold of is this. We can’t have one without the other. This means that diffraction can be observed in a double-slit interference pattern. Essentially, this is because each slit emits a diffraction pattern, and the diffraction patterns interfere with each other.

Screen Shot 3.png

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The shape of the diffraction pattern is determined by the width (W) of the slits, while the shape of the interference pattern is determined by d, the distance between the slits. If W is much larger than d, the pattern will be dominated by interference effects; if W and d are about the same size the two effects will contribute equally to the fringe pattern. Generally what you see is a fringe pattern that has missing interference fringes; these fall at places where dark fringes occur in the diffraction pattern. So, put another way, we see the broad diffraction envelope and underneath it, the equally spaced interference fringes. When an interference fringe sits underneath a diffraction minimum, we can’t see it. These so-called missing orders are a favourite in exams.

This treatment is detailed, but worth some study, as is this Java applet from Walter Fendt which just illustrates single slit diffraction. You can vary both the slit width (narrow slit = more diffraction) and the wavelength (longer wavelength [redder] = broader pattern.

Waves and wavelets

It’s all here

Huygen’s principle – the notion that a plane wave in an isotropic medium consists of a large number of adjacent circular disturbances – wavelets- which summate in space and time to make a new wavefront can be used to explain reflection, refraction, diffraction and so on.

This is a beautiful little applet which explains refraction perfectly. More to follow later.

Take a look

Reflection: Each point on a plane wavefront can be considered to be a circular ripple. The time taken for the ripple at A to get to B is the same as the time taken for the ripple at C to get to D. The speed is unchanged in the medium, so simple geometry shows that the angle of incidence equals the angle of reflection.


Refraction: Imagine a column of soldiers, marching in step on a road. Suddenly, the road becomes muddy on one side. They’ve all been told that AT ALL COSTS they have to march in step (frequency doesn’t change). The only way they can manage is that the ones in the mud march with smaller steps (shorter wavelength, lower speed). The result is that the whole line changes direction.

Light passing from a less to more optically dense material will go more slowly. The ratio of speeds (fast to slow) is called the refractive index,n, between the two materials. From air to water n=1.33

Another way of looking at refraction (thanks to Richard Feynman) is the lifeguard on the beach who has to reach the drowning swimmer in the water. He can run faster than he can swim, so what path does he have to take to reach the swimmer in the shortest possible time?

This is a neat analogy of  Fermat’s principle or the principle of least time – the idea that the path taken between two points by a either a lifeguard needing to get to a drowning swimmer or a ray of light is the path that can be traversed in the least time, which turns out to look like the standard diagram showing refraction between two media – the beach being the less ‘optically dense’ than the sea.

This is Dick Feynman’s original diagram (supposedly)

Isotopes

There are just over 100 elements, but over 1500 ISOTOPES. Atoms of the same element can have different numbers of neutrons; the different possible versions of each element are called isotopes. It’s rare in chemistry, but just for a moment, let’s forget about the electrons and concentrate on the tiny, massive nucleus in the middle.

For example, the most common isotope of hydrogen has no neutrons at all; there’s also a hydrogen isotope called deuterium, with one neutron, and another,tritium, with two neutrons. These are both found in the Sun.
 

Hydrogen   Deuterium   Tritium

If you want to refer to a certain isotope, you write it like this: AXZ. Here X is the chemical symbol for the element, Z is the atomic or proton number and A is the number of neutrons and protons combined, called the mass number. For instance, ordinary hydrogen is written 1H1, deuterium is2H1, and tritium is 3H1

How many isotopes can one element have? Can an atom have just any number of neutrons?

 

 

 

No; there are “preferred” combinations of neutrons and protons, at which the forces holding nuclei together seem to balance best. Light elements tend to have about as many neutrons as protons; heavy elements apparently need more neutrons than protons in order to stick together. Atoms with a few too many neutrons, or not quite enough, can sometimes exist for a while, but they’re unstable. 

 …not sure what you mean by “unstable.” Do atoms just fall apart if they don’t have the right number of neutrons? 

 

 

Well, yes, in a way. Unstable atoms are radioactive: their nuclei change or decay by spitting out radiation, in the form of particles or electromagnetic waves.

 

 

Carbon is a good example. Most of the carbon that we come across is carbon 12, with 6 neutrons. The isotope carbon 14 is much rarer, with 8 neutrons and is radioactive, decaying very slowly over thousands of years to nitrogen 14 with a neutron mysteriously turning into a proton and firing out a high energy electron from the nucleus (yes, that’s right) which we call a beta particle. This is the basis of carbon dating. Plants trap or ‘fix’ atmospheric carbon during photosynthesis, so the level of 14C in plants and animals when they die approximately equals the level of 14C in the atmosphere at that time. However, it decreases thereafter from radioactive decay, allowing the date of death or fixation to be estimated. It works OK for things up to about 60,000 years old, assuming the tiny percentage of carbon 14 in the atmosphere has stayed the same for this time. This site’s about carbon dating amongst other things and is quite funny. Have a look at it. It’s got some experiments to try on it as well as cartoons. ASK BEFORE YOU TRY ANYTHING AT HOME!

Energy in Simple Harmonic Motion (SHM)

This link is from a book

It’s well worth a look or two, if only to illustrate the extent of some of these applications in physics and also the fact that SH oscillators are nothing more than periodic exchangers of KE and PE.

Since KE is proportional to v² the frequency of the KE/time graph is twice that of the v/t graph. The same applies to the PE /time graph, but it is inverted. The graphs tell the story

 

if Screen Shot 3.png

then kinetic energy against time is a cosine squared function and so potential energy against time is a sine squared function, both oscillating at twice the frequency of the displacement. Think this through carefully by imagining either a pendulum or a mass-spring oscillator during various points in its cycle and remember, energy is always positive.

Screen Shot 2.png

 

For a perfect oscillator, the sum of the energies at any point in the cycle is constant, whether we plot energy against time or energy against displacement as shown in the document. Download a copy and print it for your notes. Here’s the graph – it’s parabolic, clearly because of the squared term in v.

For energy against time:

and energy against displacement

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HL IB students: you should be able to use these equations. Don’t attempt to measure gradients of graphs unless explicitly asked to do so. This way is better. Starting with this

Screen Shot 7.png

showing us that when x=r, at the extremities of the oscillation, v=0, as we’d expect. Replacing r with  x0 we can find values for velocity kinetic energy and potential energy for any value of displacement.

Screen Shot 6.png


 

Density

This is easy. Density is a property of a MATERIAL, not of an OBJECT. We can talk about the density of wood, but not the density of a table.

DENSITY = HOW MUCH STUFF {mass in g or kg} ÷ SPACE THE STUFF TAKES UP {Volume in cm³ or m³)

With regular objects: measure dry mass on a top-pan balance, record it then lxbxh =V

{Dry? If we get it wet, we’ll be measuring the mass of the water as well….}

Same deal if the objects are: Cylinders: volume = ∏r²h   Spheres: volume = 4/3∏r³

For ‘knobbly’ things, use a displacement can or dunk in a measuring cylinder.

This is a bit harder.

In the Radisson SAS, the Kugel Ball  is kept afloat on a jet of water. I estimated it to be about 1.6m in diameter. If you’re smart, you could work out its volume in cubic metres. The density of marble is about 2550kg/m³. Find out the mass of the Kugel ball in kg. And, you can move it with one hand….

 l.Kugel Ball

Here’s a list of the densities of some common materials. Have a look at it. You don’t have to remember any of the numbers…

Rules of thumb: if gases have a density of 1 unit, liquids have a density of 1000 units and solids about 10,000 units

metallic microlattice is a synthetic porous metallic material, consisting of an ultralight form of metal foam. Its creators claim it is the “lightest structural material” known, with a density as low as 0.9 mg/cm3. 

Wow…

Damped Oscillations and Q factor

 

An oscilloscope displays damped oscillations
An oscilloscope displays damped oscillations

We know that in reality, a mass on a spring  or a pendulum won’t oscillate for ever. Frictional forces will diminish the amplitude of oscillation until eventually the system comes to rest.

To remind you, here’s a mass-spring oscillator

We can add frictional forces to a mass and spring oscillator. Imagine that the mass was put in a liquid like engine oil. You won’t like cleaning the weights afterwards, but be that as it may, when the mass is inside the oil,  the amplitudes of successive oscillations will be reduced, and probably it’ll hardly oscillate at all, just bounce once, overshoot a little bit, then come to rest.

On the other hand, as you’ve probably discovered, a mass in air oscillates many times before it comes to rest. To incorporate friction, we can just say that there is an opposing frictional force that’s proportional to the velocity of the mass. This is a pretty good approximation for a body moving at a low velocity in air, or in a liquid. This is known as damping.

Screen Shot 2016-11-30 at 14.34.05.png

The spreadsheet damped-oscillations allows you to play with the amount of damping. Investigate what happens to the time period of the oscillations as the “damping factor” increases.Change the yellow cell to any value between 0 and -2 – this is a damping factor, related to Q (see below) then look at the resultant graph. The graphic above is a screenshot from it, showing moderate damping.

In a car shock absorber system, do we want the damping to be light or heavy?

Look at either of the diagrams. What is happening to amplitude with oscillation number? Have you seen a curve shaped like this before? The amplitude decays exponentially, so if you’ve seen charge decay off a capacitor, or voltage across one during discharge, or radioactive decay, they behave in a similar way mathematically. In this case,

Screen Shot 2016-11-30 at 14.44.12.png

where A= amplitude, k represents damping and n is the number of oscillations. The – sign indicates that the amplitudes decrease with increasing n.

For a single damped mass-spring system, the Q or Quality factor represents the effect of simplified viscous damping or drag where the damping force or drag force is proportional to velocity. The formula for the Q factor is:

{\displaystyle Q={\frac {\sqrt {Mk}}{D}},\,}

where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation

Screen Shot 2016-11-30 at 14.55.15.png

 

where v is the velocity.

A system with low quality factor (Q < 12) is said to be overdamped. Such a system doesn’t oscillate at all, but when displaced from its equilibrium steady-state output it returns to it by exponential decay, approaching the steady state value asymptotically.  A system with high quality factor (Q > 12) is said to be underdamped Underdamped systems combine oscillation at a specific frequency with a decay of the amplitude. Underdamped systems with a low quality factor (a little above Q12) may oscillate only once or a few times before dying out. As the quality factor increases, the relative amount of damping decreases. A high-quality bell rings with a single pure tone for a very long time after being struck. A purely oscillatory system, such as a bell that rings forever, has an infinite quality factor. A system with an intermediate quality factor (Q = 12) is said to be critically damped Like an overdamped system, the output does not oscillate, and does not overshoot its steady-state output (i.e., it approaches a steady-state asymptote). Critical damping results in the fastest response (approach to the final value) possible without overshoot. Use the spreadsheet to try to duplicate these conditions.

Wave Motion

Mechanical waves are waves which propagate through a material medium (solid, liquid, or gas) at a wave speed which depends on the elastic and inertial properties of that medium. There are two basic types of wave motion for mechanical waves: longitudinal waves and transverse waves. This is a great link and shows the difference between longitudinal and transverse wave motion. if the direction of propagation is at 90 degrees to the direction of travel, the motion is transverse (water, some earthquakes, all electromagnetic) If the oscillation direction is the same as the propagation direction, the waves are longitudinal (acoustic waves). Slinky springs can be made to do both. 

While you’re here, check out this site as well. There’s a lot here and we’ll return to some of it later.

Transverse waves: each particle executes SHM in the vertical plane, travel in the horizontal plane (oscillation perpendicular to propagation)

Longitudinal waves: each particle exercises SHM in the horizontal plane, in the same direction as travel.

With one important exception, all waves need a material medium to travel through – the exception being EM waves which can travel through free space.

Waves universally obey the wave equation

v or c = fλ

where v or c = velocity, f = frequency, and λ = wavelength

Terminal Velocity – even for Electric Cars

267 mph, 0-60 in 2.4 secs. Aluminum, Narrow Angle 8 Litre W16 Engine with 1200 hp, $2,400,000

Tom Cruise has one of these. It’s a Bugatti Veyron Super Sport, one of the fastest production cars in the world this year. It could go even quicker but the speed has been limited because the tyres won’t be able to handle it if it goes faster.

A newer contender is the Hennessy  Venom which outstrips the Bugatti and actually costs less. The Koenigsegg Agera RS is faster still, at 277mph, but, wait – the big news next year will be that Elon Musk’s Roadster will outstrip them all – an electric car able to do 0-60 in a blistering 1.9s and an as yet unannounced top speed.

Screen Shot 2016-02-19 at 11.17.37 AM
This Hennessy Venom recorded an unregistered 435km/h (270.49 mph) on the landing strip at the Kennedy Space Center in Feb 2014.

Q: Why do cars have a top speed?

A: When the forward thrust of the engine is balanced by the resistive force of air pushing back against it, all forces are in equilibrium and so there’s no change in speed.

Have you noticed, a 300bhp sports car can go faster than a 300bhp tractor, since the air slides over the sports car more efficiently so it can go faster before the air resistance is as large as that experienced by the tractor.

Skydiving
Skydiving

When a parachutist opens his chute, his weight is quickly balanced by the push of air under the canopy, so he falls at a steady speed. We recall that weight is a force. Unbalanced forces change a body’s state of motion (speed it up or slow it down. Here’s the textbook stuff.

An object falling vertically has two forces acting on it-weight acting downwards and air resistance acting upwards. At the beginning of its fall, travelling slowly, weight is larger than air resistance, so it accelerates downwards.

As speed increases, weight stays the same but air resistance increases, so the net force and consequent acceleration decreases.

When air resistance and weight are the same, net force is zero and the body falls at a steady speed. (AS students – Newton 1) This is called TERMINAL VELOCITY.

We’ve all seen modern parachutes – this image is of a da Vinci parachute which he invented in the 16th century. A Swiss man made one recently and actually used it on a 3000m drop. EEK!

Now download terminal-velocity-2and try a terminal velocity experiment yourself. My son showed me this when he went to University. I was surprised – perhaps you will be too.

A Lot of Mass and Weight

 

Failaka Island, Kuwait            

Failaka Island, Kuwait

Failaka island in Kuwait isn’t very big.

Stood shoulder-to-shoulder, however, the entire population of China (over 1 billion people) could fit on it. Neglecting the fact that the ones in the middle would be incinerated by the heat, thus combustion gases would decrease the mass, this still represents an awful lot of stuff. Which is, of course what mass is. Amount of stuff. Measured in kilograms. If the mass of one Chinese person is, on average, 50kg, this means that Failaka island would have 50 billion kg of extra mass on it. The Earth pulls each kilogram towards its centre with a force of about 10N,  – we call this the pull of gravity – so the weight of all those people would be 50 billion x 10N or 500 billion N – about the same weight as TWO HUNDRED AND NINETY STARSHIPS (if Captain Kirk’s figuring is accurate). Go ahead. Check out the link. More than enough to destroy the whole Klingon Empire….

Do you think the island would sink under all that weight?

I have a mass of about 80kg on Earth, which means I have a weight of about 800N. The pull of the Earth determines my weight, but what about the pull on other places?  Go here and fill in your weight in N. The site will tell you what you’d weigh elsewhere in our Solar System and in other places too. (If you want to do it again, you have to reload the page.) Your mass, of course, wouldn’t change, you’d still be you, even on Mars. If you had long, spindly legs like a giraffe, could you survive on Jupiter?

The little bubble in the picture is me. Shouting for help.

Fractional Distillation

A Refinery. There's a few here, but this one's in England
Refinery in Hampshire UK

..is the process whereby crude oil  – the unprocessed stuff that comes out of the ground – containing many different hydrocarbons is separated into usable fractions because they boil at different temperatures. You basically heat crude oil up, let it vaporise and then condense the vapour.

  1. Heat the crude indirectly with high pressure steam to temperatures of about 600°C.
  2. The mixture boils, forming vapour (gases); almost all substances go into the vapour phase.
  3. The vapour enters the bottom of a long column (fractional distillation column) that is filled with trays or plates.
    • The trays have many holes or bubble caps (like a loosened cap on a soda bottle) in them to allow the vapour to pass through.
    • The trays increase the contact time between the vapour and the liquids in the column.
    • The trays help to collect liquids that form at various heights in the column.
    • There is a temperature difference across the column (hot at the bottom, cool at the top).
  4. The vapour rises in the column.
  5. As the vapour rises through the trays in the column, it cools.
  6. When a substance in the vapour reaches a height where the temperature of the column is equal to that substance’s boiling point, it will condense to form a liquid. (The substance with the lowest boiling point will condense at the highest point in the column; substances with higher boiling points will condense lower in the column)
  7. The trays collect the various liquid fractions.
  8. The collected liquid fractions may:
    • pass to condensers, which cool them further, and then go to storage tanks
    • go to other areas for further chemical processing

Fractional distillation is useful for separating a mixture of substances with narrow differences in boiling points, and is the most important step in the refining process.

Refineries must further treat the fractions to remove impurities. They combine the various fractions (processed and  unprocessed) into mixtures to make desired products.  For example, different mixtures of chains can create petrol with different octane ratings.

Print out a copy of this Word file “Crude at a Glance” and learn it carefully – it gives all the detail you need to know