Sundial and marmalade at Dalemain

We went to Dalemain House near Penrith for the wonderful marmalade festival and found this lovely old sundial. It had markings on for the minutes as well as hours.

 The best bit was that it tells you the latitude (note the old-fashioned spelling).

 Which meant that I had a chance to check the thing that I discovered just before Christmas... that the angle of the slanted gnomon to the horizontal should be the same as the degrees of latitude north of the equator. I wasn't carrying a protractor so I tried to photograph it as square on as possible. On my screen, it measures as 56 degrees which suggests in real life it will be the same as latitude. I'm still thinking about what happens on the equator where the latitude is 0 degrees.

 Mrs B's first ever marmalade scored 15/20 which was a triumph. I posted a few weeks ago about the thermometer used in its production.

 

Braking distance experiment

First of all we set the ramp to a particular height measured from the floor with a metre stick.

Then we let the trolley roll down the ramp into the box.

 The trolley was stopped by friction between the box and the floor. We measured how far the box slid. We found that the higher the ramp, the faster the trolley and the further the box slid. It's like the braking distance for a car which is also due to friction.

 Then we taped a 1kg mass to the trolley. It slid further each time.

When we repeated with 2kg, it went even further. Heavy vehicles and fast vehicles have bigger braking distance.

Riemann sphere

I posted last month about Riemann surfaces. The Riemann sphere is a simple Riemann surface. I made a model of it to help me to understand it.
First, a quick recap. You can't get an answer for the square root of a negative number. Try it on your calculator for square root of -1. You get "Math ERROR". But someone was brave enough to break the rules and pretend that the answer existed. So the answer was called an IMAGINARY number and square root of -1 is called i. i stands for imaginary.
You can then make up COMPLEX numbers which include a real number and an imaginary number eg 1 + i or 6 + 4i. Someone then suggested plotting these like you would plot x and y on a graph. The x-axis gets the real number and the y-axis gets the imaginary number. The graph is called the complex plane. A Riemann sphere is a technique for turning a flat piece of complex plane graph paper into the curved surface of a sphere instead. Here's a picture of a flat complex plane with a glass sphere above. I have placed a light bulb at the north pole of the sphere. I marked an x on the glass. The light shines through and makes a shadow on the flat complex plane. These two points are the same - it's just one is plotted on a flat surface and the other on a curved surface.

Next I drew a line round the equator of the glass. I went over the shadow on the flat surface with a red pen. We call this the "unit circle" on the complex plane. The circle has a radius of one unit. Points drawn below the equator on the glass appear inside the unit circle. They are less than 1 - in other words they are fraction. If I use the letter z to represent any complex number, the points below the equator must be 1/z to be fractions.

Next I did the thing the other way round. I marked a point 1 + i on the flat surface and worked out using shadows where the same point would appear on the curved surface. Notice that it is above the equator.

The further up the sphere you mark a point, the further from the origin of the graph the shadow appears on the flat graph. If you think about it, by the way that light makes shadows, two points close together near the top of the glass sphere will end up a long way apart on the flat plane. The conclusion is that the light must have been placed at a point representing infinity. The beauty of the Riemann sphere is that it takes a infinitely big flat plane and maps it onto a finite surface of a sphere. An infinity can be represented by a single point at the north pole.

Now I understand the construction, I need to know what it is used for! Back to Roger Penrose's book! And many thanks to my technicians for coming up with the perfect spherical flask for the photographs!

Derwent Water as a ripple tank

 I use a ripple tank to show the properties of waves to my classes. Today Derwent Water was acting as a ripple tank. The waves were coming in towards me as I took the picture. The straight edged waves appearing in the picture would be called plane waves. They hit the smaller rocks close to the shore and that produced circular ripples that were reflections. Reflection should produce waves with the same wavelength but if you enlarge the photograph above, you'll see a disturbance in the water between the nearest rock and the National Trust Centenary memorial. There was a small rock coming in and out of the water. The circular ripples it produced had a smaller wavelength than the plane waves coming in. There can be two reasons for this. One is that there is a refraction effect and that the waves are slowing down in shallower water around the submerged rock. I saw no evidence of the plane waves bending though. The second reason would be that the rock was being hit at a higher frequency than the way the plane waves were formed. This might be the answer because of the way that the rock was sometimes submerged and sometimes out of the water.

The bigger rock in this picture was never submerged and seemed to produce circular ripples with a wavelength closer to that of the plane waves.

Have I proved Newton wrong?

 I set out to investigate Newton's 2nd Law using this apparatus. The light gates were connected to a computer to calculate the acceleration. All I had to do was to sort out the forces. I had the trolley pulled along the ramp using weights hanging on the end of a string. The trouble is that as the trolley rolls along the ramp there is friction which acts as a counter force. The overall force on the trolley, the RESULTANT FORCE is weight - friction. I only wanted the weight: I needed to get rid of friction. I decided to cheat as shown in the picture below. I propped the ramp up with books so that the extra bit of gravitational pull cancelled out the friction. Or so I thought! (This method is called FRICTION COMPENSATION)

Here are my results:

The force is in the left hand column. I used a mass of 2kg so 2kg multiplied by the acceleration should equal the force according to Newton's 2nd Law. That clearly hasn't worked! Have I proved Newton wrong? I won't be sending a paper to PRL anytime soon... I must just have got my friction compensation wrong.

Efficiency question: the lock at Aldermaston Wharf

 There aren't any canal locks in north Cumbria. This one is in Berkshire. Canals were waterways made by people about 250 years ago to move large amounts of heavy stuff like coal. They were superseded by railways. How do you get a boat to go uphill? That's where the lock comes in. Start with the top picture. They open the gates and sail the boat in. They close the gates behind them. Now look at the picture below. Small panels in the bottom of the gate are opened and water pours in. This raises the boat to the top level. The gates are opened and the boat sails off.

 1. The boats are called narrowboats like the black one in the picture. I have read that their mass is 1000kg for every metre of length. How heavy is the narrowboat?
2. The narrowboat gains gravitational potential energy mxgxh by being pushed upwards by the water pouring into the lock. Calculate the gravitational potential energy gained by the boat (say g=10 N/kg because we are estimating quantities)

 3. The energy to raise the boat comes from the water that falls into the lock from the higher section of the canal. That water loses gravitational potential energy. Calculate the mass of water that falls into the lock by estimating the volume of water needed to fill the lock up to the higher level and then using the fact that 1 cubic metre of water has a mass of 1000kg.
4. Next calculate the potential energy lost by the water as it goes in by using mxgxh.

5. Calculate the efficiency of the lock by working out what percentage of the gravitational potential energy lost by the water is given to the boat.

Souther Fell - no sign of the spectral army

Wainwright's guide to the Northern Fells reports that in 1745, an army with horsedrawn carriages was seen along the top of this hill. It took 2 hours to pass and seemed to disappear over a cliff. If you Google it, you'll find that it happened more than once. They went up to look for evidence and found no trace in the marshy ground. An army of ghosts, then: spectres.Wainwright suggests that it could have been a mirage. Mirages are due to refraction when different layers of air have different temperatures. The different temperatures mean that each layer has a different refractive index. Colder air has the higher refractive index, thus light bends towards the normal line as it passes into colder air. The classic desert mirage has the hottest air nearest the ground due to the heat of the desert sands. The light can experience total internal reflection as it comes down from the colder air above and thus bend back up towards the observer. Your brain thinks light has gone straight so it appears to you that an object is below you when it truth it is above. Yes there are clearly different temperature bands as you go up a mountain. It was Midsummer Day when the spectral army was seen which might warm the atmosphere more. But I've not experienced any mirages in my years in the mountains. We saw no ghosts.

Black flask and warmer batteries

I managed to get an extra day out of the batteries that seemed to have run out in the cold (see previous post). I kept the batteries in my pocket to keep them warm and was able to take this picture and another 30 besides! This flask interested me. Mine is a lovely shiny silver. Physics teachers are fond of pointing out that if hot objects are shiny silver, they are poor emitters of infra-red heat radiation whereas dull black surfaces are good emitters. I think an experiment is called for here to see if the black outer colouring does impair the performance. It is quite possible that so little thermal energy is expected to reach the outer wall in the time you'd want to drink that the outer colour is irrelevant.

Cold batteries - rate of reaction

A lovely view of the Scafell group from High Raise. But I was down to my last set of batteries. Batteries don't perform as well at low temperatures because they rely on chemical reactions to release electrons. The flow of electrons are the current supplied by the battery. At low temperatures, the molecules have less kinetic energy and move more slowly. They collide less often and with less energy so fewer reactions can happen. Too few electrons are released to operate the camera. Oh well.

Half life of the head on a beer

After the half life of chocolates triumph last October, today we measured the half life of the head on a beer. I'll never get a job behind a bar...

 We measured the height of the froth every 20 seconds.

We plotted Ln(height/mm) against time. I've edited the results for the first 200 seconds because the results deviated from a straight line after that time. So the decay is exponential only at first.

We showed that the gradient is -decay constant. Half life = Ln2/decay constant, which gave a value of 150 seconds. Other results were around 120 seconds.

Yield and dynamic equilibrium

I photographed this helium balloon at my cousin's 40th wedding anniversary last summer. I can remember the wedding. Must be getting old... Anyway, it reminded me of an important rule in science about gases. Equal volumes of different gases always contain the same number of gas molecules.
Now consider this reaction:

All 3 of these chemicals are gases. Now, there is 3 hydrogen molecules for every nitrogen molecule, so hydrogen gas will have a volume 3 times the nitrogen. In fact, there are 4 molecules on the left hand side of the equation (one molecule of nitrogen and three molecules of hydrogen) but there are only two molecules on the right hand side. So if we are making ammonia, the reactants will take up twice as much space as the product. If we increase the pressure around the reaction, there will be less space available and that will favour the forward reaction that makes ammonia, because in the end the ammonia takes up less space than the nitrogen and hydrogen. We can also make the reaction go faster by heating it up, but high temperature tend to make the ammonia fall apart back into nitrogen and hydrogen. This reaction is complicated because you can make ammonia but it can also fall apart again. So we need to choose conditions that make as much ammonia as quickly as possible with as little as possible falling apart, and we need to do it as cheaply as possible.

Calibration - a useless thermometer?

Look at the thermometers pictured below. They have no scale on them. So we set out to add scales. We started by putting them into an ice-water mixture and marking the level on the glass with a waterproof pen. The temperature of melting ice stays fixed at 0 degrees Celsius.

Then we put the thermometers into water that was boiling vigorously. We were able to mark 100 degrees Celsius.

The last stage is to measure out an even scale between 0 and 100. In our case, every 10 degrees Celsius would have been appropriate. This process of using repeatable fixed points to set up a scale is called calibration. Extra details would be to note that we should use distilled water. Impurities lower the melting point and raise the boiling point. The experiment also needs to be done near sea level because water boils more easily in reduced pressure at altitude. But that's another story...

Observation question #3

On previous occasions, I have asked you to make an observation. This time I'm making the observation and asking you to explain it. I took this photograph in the Pennines last weekend and noticed that there was far more snow on the left side of the wall than on the right side, even though they must get the same amount of sunlight. Explain why. (The darker vegetation is heather)

The Milky Way's Magnetic Field

I liked this picture: http://apod.nasa.gov/apod/ap150127.html It's interesting that the Milky Way has a magnetic field. You get a magnetic field around a current carrying wire because of the movement of the charges within the wire. In this case, it isn't electrons that are moving - it is charged gas atoms. These ions are being rotated around the galactic centre as the Milky Way turns and thus generate a field.

The Great Circle - Flights to America

 We climbed up to High Cup Nick in the Pennines, Here's the view looking south onto a cloud inversion around the Howgills. can you spot the vapour trail in the middle of the picture? It's a truism in these parts that these planes are going to America because the shortest route is up and over. It's called the Great Circle. I got out my atlas and traced out the route from London to San Francisco - see below. On the flat map it would go west over Bristol.

 But then I got my globe out. On a curved suface, the shortest route will be different. It's easy to do by pulling a string tight between two points as shown below for the same route.  A Great Circle is drawn between two points on the surface of the sphere and is part of a continuous circle round the globe centred on the middle of the Earth. It's like an orbit in that regard.

                                       
 In close up, it does indeed go up and over Cumbria.

I used this popular website to test my theory http://www.flightradar24.com/ but I found that rather a lot of the flights over Cumbria are actually to Glasgow! There must be more of those than flights to LA. The globe showed me that the quickest way to New York is actually out over Bristol.

Skiddaw by moonlight

We climbed up to the Watches above Bassenthwaite by moonlight and I took this lovely picture of Skiddaw. Well, I could see Skiddaw clearly but it hasn't appeared in the photograph. You can see the tip of my ice axe reflecting the moonlight. The CCD pixels in my camera are not as sensitive to light as the rod cells in my eyes. Rod cells are found round the edges of your eyes and cannot detect colour. The pigment in them is depleted in bright light but regenerates when it goes dark - hence the dark adapted eye. A rod cell can detect a single photon of light which makes it up to 100 times more sensitive than the cone cells in the middle of your retina. Also, more than one rod cell is connected to the same nerve fibre so it can amplify the signal. This does reduce the resolution of the image. They also record photons for a longer time - it takes about 0.2 seconds to build up an image. This makes the picture seem brighter but makes it harder to see very sudden movements. Being around the edge of your retina, it explains why the path seems bright until you look down at it directly.

Measuring tyre pressure

I was sorting out the pressure on my van tyres. Pressure is out of fashion as a topic in school Physics at the moment. Pressure is force per unit area, and thus it would be measured in Newtons per square metre.

 The reading above is in the old Imperial units of pounds per square inch. They are not really dimensionally correct as units because pounds is a measure of mass not force. But it does give a good mental picture of what's happening. That's about 30 bags of sugar balanced on a square inch - say on the area of a couple of 2p pieces.

 Bar as a unit of pressure was introduced just over 100 years ago. 1 bar is roughly atmospheric pressure. 1 bar = 1000 millibars so air pressure in weather forecast is often given in millibars. A low pressure system might have a pressure of 980 mb, so not much below normal atmospheric pressure. My tyres were 4 1/4 times atmospheric pressure.

 And finally to proper Physics units. KiloPascals. 1 Pa is 1 Newton per square metre. As you can work out, 1 bar = 100,000 Pa.

 

An experiment involving Fleming's Left Hand Rule

As it happens, today we did an experiment which can be explained in part by Fleming's Left Hand Rule. We clamped a wire to the table and tensioned it with masses hanging over the edge of the table. We wired it up so that alternating current was flowing through the wire. The wire passed between the poles of a strong magnet. Hence we had UVW from yesterday's post - cause, connection and effect. The current is the cause, the magnetic field the connection and the force (movement) is the effect. The current keeps changing direction but the magnetic field is fixed. Hence the force also alternates and the wire oscillates. If the frequency of oscillation is such that half a wavelength fits between the two wooden wedges, you get a "one loop" stationary wave - the fundamental mode. Changing the mass affects the tension in the wire and thus how easily it can vibrate. Hence the length of the stationary wave loop changes.

3-Finger-Regel (UVW-Regel)

I was given this wonderful mug with Physics equations on it. I was interested in Fleming's Left Hand Rule. The problem is that it is actually the RIGHT hand. In England, we use the right hand for the generator rule. The left hand is called the motor rule because it is about putting a current in and getting movement out. This is definitely the same rule because of the UVW mnemonic. Ursache is the cause and Wirkung is the effect. Technische Strom is conventional current which goes into the motor, so it is the cause. Kraft is the force - the effect. Richtung means the direction and, if you were wondering, Vermittlung can be translated as "connection". Here's the way I use Fleming's Left Hand Rule:

All that has happened is that the German rule swaps force with current and if you use right hand instead of left, magnetic field will point in the same direction.

Sugar thermometer - making marmalade for Dalemain

Mrs B was making marmalade for a competition at the Dalemain Marmalade Festival http://www.dalemainmarmaladeawards.co.uk/ She was using a specialist "sugar thermometer" because the sugary liquid boils at well above 100 degrees Celsius. We don't tend to use mercury thermometers anymore. Digital is the order of the day. These old-fashioned ones work by having a liquid like mercury inside a narrow capillary tube. As the liquid is heated it expands and the only place it can go is up the tube. If you look at the scale you will see that the expansion is linear. The scale on the right is Celsius and that on the right is in Fahrenheit. We tend to teach that the precision is the smallest scale division - in this case + or - 2 degrees Celsius. But most physicists argue that it should be half that because you can tell where you are in the gap. We could increase the precision by making the tube narrower because there would be the same volumetric expansion with a smaller cross-sectional area. This would make the scale divisions wider and often the chance to fill in finer scale division lines. A scale like this has to be calibrated first. It is normal to fix 0 degrees Celsius with melting ice/water mixture and 100 degrees Celsius with boiling water. But only the latter appears on this scale so I wonder what they use for the second point.

A question of balance

 Below is a picture of an old-fashioned amplifier that I received as an 18th birthday present! This article is about the balance control on the right hand side.

An exam question we did recently related to the balance control on a stereo. In the days before ipods, there used to be chunky bits of equipment that sat in the corner of the room connected to two different loudspeakers. It was called stereo because they record different tracks, one for each speaker. This continues, of course, but you'll notice it as different tracks to left and right headphones. The problem was that if you sat in one part of the room, you'd be closer to the right speaker than the left and so its track would be louder. The balance control allows you to turn up the volume of one whilst turning down the volume of the other. The exam question concerned the wiring. Look at my top diagram. Both speakers are controlled by individual variable resistors. Hence you can turn each speaker up or down individually. But you'd need two control knobs. The second circuit with a single potentiometer shows how it is actually wired. I've written on the potential differences assuming it is a linear potentiometer. You only need one control knob.