Analysing some knots

I've been reading more of Matt Parker's book on maths. I wasn't aware that Lord Kelvin had proposed the idea that matter consisted of knots in the ether, at a time before the existence of atoms was proven and when it was still believed that the ether permeated the whole universe. The idea was that each element was a different knot. So the study of knots was started. I've done my first, shown above. It is the simplest possible knot but not as simple as  thought when loosened slightly. It's called a trefoil knot and has 3 crossovers. More to follow!

Amazing conglomerate rock on Rhossili Down

This rock on the top of Rhossili Down looks like concrete but it's natural. It is evidence of the recycling of rocks during the Earth's history. The rock itself is old enough to have been thrown up to make the hills overlooking the sea but the pebbles within it come from much older rocks that had been weathered and transported. They are round so they must have been carried some distance. They say that the pebbles were deposited by a river that flowed across this landscape millions of years ago.

Solving the Wobbler

Here's the solution to the Wobbler problem. It's not a universal solution because I've just used the two positions shown in yesterday's pictures. The Wobbler is made from two cardboard circles intersecting. In my solution, I have taken the slot to be a x the radius, where is a factor, a fraction of the radius.

Making a Wobbler

I have been reading this wonderful book. Matt Parker is comedian and mathematician. I realised that given the title of the book, I should actually make the things he suggests. My problem has usually been that I have read and understood the theory but never tested it. Perhaps I'd be a real physicist if I'd tried things earlier!

 I made the Wobbler from page 65. It's two intersecting circles like this. The idea is that it rolls like a wheel because if the size of the slot is exact, then it will continue to roll like a wheel because its centre of gravity remains the same height above the ground.

 By considering the two positions shown below, I was able to prove his statement that the exact size of the slot is 29.2893% of the radius. Details to follow!

Sand ripples on the beach at Rhossili

It was suggested to me a couple of weeks ago that the ripple marks in the altostratus clouds that have attracted my attention are formed in the same way as ripple marks in the sand. I got thinking about it again on the beach at Rhossili. First we crossed a fresh water stream flowing into the sea. The water flowed over some pebbles and then formed the characteristic ripple pattern shown in the top photograph. The sand on the bottom of the stream was likewise rippled. It occured to me that the pebbles had probably made the flow of the water turn turbulent. There is certainly turbulence in sea water - witness the white water. There are the competing flows of water in and out. But why only ripple marks in some places? An initial look at Internet sources suggests I've got some more observations to make - perhaps measuring the wavelengths of the ripples and looking at the cross-sections. My current working hypothesis for the altostratus cloud is going to be turbulent airflow.

Wind power for the Worm's Head at Rhossili

 The picture above shows Worm's Head, a tidal island at Rhossili on the Gower. The coastguard station below had a small wind turbine. The wind speed was said to be about 20mph average that day, a stiff breeze.

Conventional wisdom is to assume that the air stops dead when it hits the blades and to assume that all of the kinetic energy of the air is transferred to the blades. Let's estimate that the diameter is 50cm. 20mph = 32 km/h (5 miles is 8 km). 32 km/h = 32 x1000/3600 = 8.9 m/s. So in 1 second, a volume of pi x 0.25squared x 8.9m is stopped. That's 1.7 cubic metres of air. Let's go with air having a density around 1kg per cubic metre. Going for 1 significant figure, then 2kg of air is stopped every second. Kinetic energy = 1/2 x mass x speed squared = 81J. Let's say the turbine is 30% efficient. That works out at no more than 30W.

Preparing for L6 Physics #8: Stationary waves on my bass guitar

I'm learning bass guitar. Like all stringed instruments, the strings are fixed at each end. When you pluck the string, you create a wave on the string that travels along the string and reflects off the far fixed it. It crashes into the bits of the wave that have not yet arrived. If you get two waves travelling in opposite directions with the same frequency, speed and amplitude, then you get a STATIONARY WAVE. The peak doesn't move along. The peak in this case is in the middle of the string and it just goes up and down in that skipping rope pattern. Fixed points are called NODES and this oscillating maximum in the middle is called an ANTI-NODE.

Brown seaweed and Bass Rock

Here's the view from the coast near Dunbar looking towards Bass Rock. It's low tide. The seaweed that has been revealed is reddy brown. It reminded me of a question on the SEG 1991 GCSE Biology paper, still my only GCSE qualification.

The top graph shows that seawater absorbs most at the red and blue ends of the spectrum. Greens and yellow will penetrate water better. Chlorophyll absorbs red and blue best but reflects green. It wouldn't be much good for plants in reasonably deep seawater. So these seaweeds have a different light absorbing pigment called phycoerythrin. The graph shows that this absorbs greens and yellows well, but not reds and blues. It is this pigment that colours the seaweed. It will look reddish because it reflects red but absorbs greens and yellows. I can describe this from the data but at the moment I can't explain it in terms of photons and electron energy levels. More on that to come. I use this question as a warning that rote learning will not get you top exam grades, though it is necessary to get the basics sorted. No one can learn every pigment in every plant. The aim here was to be able to analyse data. I came out of this exam knowing more than when I went in. That's a good exam.

Preparing for L6 Physics #7: Three quarks for Muster Mark

Finnegan's wake is reputed to be the most difficult book in English Literature (though James Joyce was Irish). I once got through 200 pages and counted detect a shred of plot. It's beautiful and poetic, with made up spellings to help evoke Irish accents. Murray Gell-Mann is a brilliant American physicist. He made an important deduction that protons and neutrons are not singular, fundamental particles but are in fact each made of 3 smaller particles inside. He'd been reading Finnegan's Wake (go figure!) and had found a quotation "Three quarks for Muster Mark". It chimed with his idea of 3 particles so he called them QUARKS. At least, that's the legend. Particles made from quarks are called HADRONS. Particles made from 3 quarks are called BARYONS (eg protons and neutrons) but there are also particles made from quark and anti-quark pairs. They are called MESONS.

A stitch in time

It was Mrs B's birthday so I made her some bunting. The first time I've used a sewing machine in four decades. This is what happens when you watch the Sewing Bee. It brought to mind a saying that I struggled to understand as a child: "A stitch in time saves nine." It took means years to realise that "in time" meant "in the nick of time" rather than "in the fabric of time itself". So I used to imagine there being a stitch somehow in the minutes and hours. First: what is time, actually? One physics teacher told me that it is that quantity which allows two objects to occupy the same spacial coordinates... think about it! Time for Einstein was just another dimension that could be stretched. That takes some thinking about. I've been trying to get my maths up to General Relativity standard recently so I'll let you know. But there is a problem trying to reconcile gravitational theories with quantum theory. When things get very small, quantum effects take over. The uncertainty principle means that eventually things are so close together that you can't tell whether you have two locations or one. This tiny length is called the Planck Length. The time it takes light to cross this distance is called the Planck Time. This may well be the smallest unit of time and that you can't meaningfully have smaller divisions. If so, then perhaps this is my stitch in time.

Preparing for L6 Physics #6: Nuclear forces

The building in the distance on the right of centre is Torness nuclear power station near Edinburgh. Energy is released from the nucleus of an atom by the process called nuclear fission. An understanding of the forces in the nucleus helps. When I was at school, it never stopped to wonder how protons could all stay together in the nucleus. They should repel. The first people to ask this question came up with a simple answer: there must be another attractive force acting that is stronger than the electrostatic repulsion. It is called the STRONG NUCLEAR FORCE because it is strong and acts between protons and neutrons. Protons and neutrons are called HADRONS. They are made of smaller bits inside. There is more detail awaiting a further post. Electrons are examples of another type of particle called LEPTONS which can't be split down any further. They don't experience the strong nuclear force. However, there is another force called the WEAK NUCLEAR FORCE that affects both hadrons and leptons.

Dolostone on the fault near Barns Ness

This is the shore at Barns Ness near Dunbar on the Edinburgh coastline. It has an impressive display of sedimentary rocks including one of the biggest limestone outcrops in the Scottish lowlands. We used a geology leaflet to find our way around http://www.edinburghgeolsoc.org/downloads/rigsleaflet_barnsnessa4.pdf 

In picture below, there is a crack in the rock in the middle of the foreground which lines up with the lumps of rock in the distance. It's a geological fault where earth movements have split the rock.

 The rocks below are examples of breccias. The original rock is fractured by the earth movements. This creates cracks. Mineral bearing liquids flow into the cracks. The water evaporates and the minerals crystalise out. The edges of the broken fragments are still sharp and jagged, showing that there was no transportation to cause erosion. In other words, they are still in the place that the fracture took place.

The leaflet says that this is dolostone. This will have formed because the inflowing mineral had more magnesium in it that normal limestone and so changed the chemical composition. I think that the dolostone must be harder than ordinary limestone because it has remained as the big lumps sticking out marking the faultline clearly.

Smoke jack at Wordsworth House, Cockermouth

This ingenious piece of equipment is in the kitchen of Wordsworth House at Cockermouth. The fire causes air to expand, become less dense and rise up the chimney. In other words, it gives kinetic energy to the air. Part way up the chimney, there is a small horizontal windmill. The rising hot air turns this. It drives the gearing mechanism shown in the top two pictures which then turns the roasting spit at the bottom to make sure that the mean is evenly roasted by the fire. So wasted energy from the fire is used to drive the mechanism. It makes the fire much more efficient.

Interference fringes at Fickle Steps

This is a photograph of the water flowing between two of the stones on the Fickle Steps in the Duddon Valley. The stripey pattern is a permanent feature - so must be some kind of stationary wave. The lines on the water look like double slit interference fringes. I've been thinking about how they form. The flow above the stones is laminar. There were no surface waves visible. The water is squeezed between the rocks. Thus some of the laminar flow is directly into a rock. There will be reflection due to Newton's Third Law: the rock stops the flow of water by applying a force. The reaction force pushed the water back, hence ripples. But why the permanent feature? It looks interference but what are the reflected waves interfering with?

Pictures of Pluto

Take a look at this http://www.bbc.co.uk/news/science-environment-33491454 I find it unbelievable that there are quality images of Pluto. When I started teaching, a picture of the whole of Pluto was about 10 pixels. And there are better photographs to come!

Putting my finger in a beaker of water

I had a lovely night out with local Physics teachers.We were set a problem: if a beaker of water is placed at either end of a balanced beam and you put your finger in one end, what happens? I found this version of the experiment described on a website and decided to try it myself. Putting my finger into the water makes the measured mass on the scales increase even though I am not touching the beaker. There are two possible explanations:
1. Adding the finger displaces water. By Archimedes' Principle, there is an upthrust on the finger equal to the weight of water displaced. In other words, a new force has been applied to the system. By Newton's Third Law, an equal and opposite force must act. The upthrust acted from the water on the finger. This new reaction force acts force from the finger on the water. Water is not compressible so the force is transmitted to the bottom of the beaker and pushes it down onto the scales. By this logic, the increased force should be equal to the upthrust and thus to the weight of water displaced. The extra mass is 5.53 grams. Water has a density of 1 gram per cubic cm. This means that the displaced water would have a volume of 5.53 cubic centimetres. I don't think that my finger is quite that big but it's not far off.
2. The second solution is that deeper water exerts a bigger pressure at the bottom. Displacing the water makes it deeper.

The effect would not be noticeable in a beaker of air for two reasons. Air is much less dense than water so the weight if displaced air is tiny and thus might not register on the scales, Secondly, air is compressible so the reaction force would probably just push the air molecules closer together rather than push down on the bottom of the beaker.

Preparing for L6 Physics #5: Transverse progressive waves

Click on the picture for the full panorama. These waves on the sea at Silloth are transverse progressive waves. They are transverse because the direction of oscillation (up and down) is at 90 degrees to the direction of energy propagation. It is a progressive wave because the crests move towards the shore. Progressive waves can transfer energy. NB If you look in detail, the motion of waves on the sea is perhaps a bit more complicated than I'm saying. Instead of just up and down, there is a rolling circular motion in the wave but there can't be many people who would decline to classify them as transverse.

High tidal range at St Peter Port

We were astonished by the tidal range on Guernsey. Admittedly, we arrived at full moon so there was a spring tide. But look at the height of the poles that allow the floating jetty to move up and down. It's something like 9m. They said the 3rd biggest in the world. Not true, but in the top fifty. See http://www.co-ops.nos.noaa.gov/faq2.html#26 I have read that they are all close to big bodies of water. True of Canada and Chile, but why not the coasts of Africa? And why the Bristol Channel but not the Outer Hebrides or Norway? Is it just about the shape of the coastal shelf?

Kites on Silecroft beach

We flew two kites of the same design on Silecroft beach near Millom. The one in the bottom left hand corner is about twice the size of the higher one. So what has affected the balance of forces? Three forces act: weight, lift and tension. The wind was almost horizontal. Is it just the greater weight of the bigger kite? But it has a bigger area and so catches more wind. Perhaps the weight pulls it down so much that the kite is almost vertical so the wind pushes it sideways rather than acting as lift.

Foghorn on the Mull of Galloway

There is a foghorn on the slope below the lighthouse, shown in the lower photograph. Light is scattered by fog so it is not possible to see the light beam in fog. But sound can travel through fog. Sound is a longitudinal mechanical wave. Air particles oscillate backwards and forwards. There is a certain amount of viscosity is any medium. This opposes the motion of the particles. Hence the longitudinal oscillations are slightly damped at each point and energy is dissipated. In other words, the perceived volume will decrease. If there were no viscosity, it would be an inverse square law like light. So the volume of sound heard must decrease faster than this. It turns out that such attenuation is dependent on frequency. The lower the frequency, the less the attenuation. Lower notes should travel further for the same initial volume - so foghorns were always low notes. I say were because they've all been turned off in the UK! See https://en.wikibooks.org/wiki/Engineering_Acoustics/Outdoor_Sound_Propagation for some of the theory.

On the Meridian in Patrington

We came across this on the roadside east of Hull. Fixing east and west was important for the modern world because it allowed the creation of time zones. Unlike the hemispheres divided by the Equator, the Prime Meridian is an entirely human creation. It could have been placed anywhere. The vagaries of history meant that the British fixed it. It led to Greenwich Mean Time, fixing 12 noon as the time when the Sun is highest in the sky at all places on this line. That was then applied to the whole country even though the Sun is highest in the sky a little later here. It's been a practical outworking of Physics that has enabled things like timetables to work on a national scale.

Preparing for Lower Sixth Physics #4: Huygens' construction in Portpatrick harbour

Looking down over the side of Portpatrick harbour, I noticed that the plane waves hitting the sides of the harbour were getting reflected back as semi-circular ripples. There must have been bits sticking out a little bit further than usual, but it reminded me of Huygens' construction. Christiaan Huygens was a Dutch physicist who was a contemporary of Isaac Newton. He suggested that light was a wave: Newton held that light was made of small particles called corpuscles. You can explain reflection and refraction with either waves or particles. It took Thomas Young's double slit experiment 200 years later to tip the balance in favour of the wave theory. Another different between Newton and Huygens was that Newton favoured using rays - arrows to show the direction in which the wave is moving. You can't see these in my photograph. You can see the crests of the waves. These are called wavefronts. Huygens' construction uses a technique called "secondary wavelets" to predict where the next wavefront will appear. He split the wave into individual points which then made their own semi-circular ripples like the ones I saw in the harbour, Then he drew the next wavefront along the front of the secondary wavelets. See below:

I-beam at Ravenglass

I-shaped beams are very common. I did study them once but have forgotten a lot. I think the idea is to have a stripped down shape that would weigh a lot less than a solid steel beam whilst retaining most of the strength. The top and bottom bits are called flanges. These make it harder to flex the beam. Here it is stretching out under the load of the railway track. They help to stop it bowing in the middle. The web in the middle resists shear forces.

Preparing for L6 Physics: Diffraction at Portpatrick

Waves are an important topic. Here straight lines of waves were approaching the harbour entrance. They are called plane waves. Diffraction is when waves bend when they go through a narrow gap. You can just about tell that this has happened in the bottom picture. For diffraction to be really pronounced, the wavelength must be equal to the size of the gap. Here the waves were maybe 1 metre apart but the harbour entrance was at least 20 metres across. Hence diffraction was not that good.