What's the difference between speed and velocity?

I got a rubber and tied it to the end of a piece of string. I held it over the middle of a dinner plate that I had put on the floor.

 Then I started to spin the the string in my hand gently so that the rubber went round in a circle, tracing out the edge of the dinner plate.

You really should do this if you can to see what happens. But if you can't find any string, take my word for it that the rubber goes round at at steady speed. It doesn't get faster and it doesn't get slower (though it will eventually run out of energy after a while).
Now I'm going to tell you that the rubber is ACCELERATING. Think about it: I just said that the experiment shows that the rubber doesn't get faster and doesn't get slower. It stays at a steady speed. But I said it was accelerating. Accelerating usually means getting faster. Now we need to remember VECTORS and SCALARS. Speed is a scalar; it is just a size (magnitude). But velocity is a vector; it has both size and direction. The rubber might not be changing speed (size) but by going round in a circle, it is changing direction. So speed stays the same but velocity changes. Acceleration is a vector too. It can mean a change in size (speed increases) or it can mean a change in direction - or it could mean both at once.
The direction changes in this example because the string is constantly pulling the rubber round into a circle. The string provides a resultant force to make it accelerate by changing direction. If you cut the string, the the resultant force vanishes and the rubber stops changing direction. It flies off in a straight line until it hits the floor. Try it!

Measuring the masts

You can see the masts at Anthorn from our back garden. They are as tall as my little finger's width when I hold it at arm's length. That means that they subtend 1 degree. In radians that is 0.017 rad. Using radians I can model the height of the mast as an arc length. Angle in radians = arc length/radius. In this case the radius means the distance to the masts, which are about 10 miles from here or 16 km. That means the height of the masts is 16000 x 0.017 = 280 metres. https://en.wikipedia.org/wiki/Anthorn_Radio_Station gives the mast height as 228 metres so that's not bad!

Coanda Effect in the washing up

I was washing up the demijohn after racking off the rhubarb wine when I noticed that the jet from the tap was sticking to the inner wall of the glass. It seemed strange to see it divert from falling straight to divert outwards. It turned out to be a useful way to clean off the yeast residue at the top. It looked like the Coanda Effect, although being no expert, another potential explanation that occurred to me could include surface tension. Coanda Effect seems to be about the flowing liquid pulling a layer of air along with it. This creates low pressure which encourages more air to flow in at 90 degrees to the flow of the liquid. Normally the same thing happens on both sides of the liquid so there is zero resultant force and the liquid falls straight. But if the liquid runs along a surface on one side, there is only the air on the other side pushing on it which knocks the flow out of line.

Trying to measure the resistivity of granite

Having noted that granite feels cold ("stone cold") so must conduct heat quite well, I decided to see whether I could measure the resistivity and thus the conductivity of the granite. It didn't work

I taped one lead to the bottom of the worktop and the other lead to the top. The reading was off the scale and thus larger 2 million Ohms. I tried the connections taped together on the top and got the same thing

I did check that it wasn't a broken lead by clipping the leads together. The leads have a resistance of a couple of Ohms. A quick search suggests that granite can have very variable electrical properties. I suppose that reflects the different compositions, even in Cumbria. I found one record of granite having a resistivity between 25 and 1500 Ohmmetres. The work top have an area of about 2 square metres and a thickness of 0.03 metres. That would give a maximum resistance of about 20 Ohms. Perhaps the contacts were poor.

Granite feels really cold so it must have a high thermal conductivity. The value I found says from about the same as yesterday's slate up to 4.0 W/m K. This felt colder than the slate which means maybe at the higher end. I tried to find out why stone conducts heat quite well but found nothing obvious. It may be that the substances that don't feel as cold benefit from trapped air (eg carpet). If there were a lot of free electrons, it would conduct heat well. I will have to try to measure the conductivity.

Specific heat capacity and thermal conductivity of slate

Painted stones have made life in Wigton happier during lockdown. Mrs B painted this slate in the greenhouse on a sunny day. When I moved it, I noticed that not only was the top surface hot but the bottom was too. The surface of slate is dark and absorbs heat radiation well. It has a specific heat capacity of 760 J/kg K https://www.engineeringtoolbox.com/specific-heat-solids-d_154.html This is the same as some of the metals I test in the lab so it certainly heats up quickly. The thermal conductivity is also quite good at 2 W/m K https://www.engineeringtoolbox.com/thermal-conductivity-d_429.html I had fun looking at other building materials for comparison. Brick is less good at conducting. I waited for another sunny day and went to find out if slates deeper down the pile, and out of direct sunlight, would feel warm. this was the case, showing that thermal energy was able to conduct quite a long way.

A high voltage humidity detector

Lockdown exercise has taken me right under the pylons. The air has been remarkably dry for 2 months now but it is noticeable on the days it has been close to rain that I can hear a crackling noise. The air must be conducting a little better.

Speed of water waves experiment

For this experiment you need a rectangular box about 30 cm long that won't leak if you put 0.5 cm depth of water in the bottom. I had a look round and found these. they include a rectangular washing up bowl (a round one won't work), a rectangular tray, a storage box, a biscuit tin (even the lid would work) and a cake tub.

However, I chose to use a baking tray. First job is to measure the length of the long side on the INSIDE of the tray. It was 33.9 cm long.

 I chose to write it in cm because the ruler is marked in cm. (Proper scientists might prefer 339 mm) But notice that I wrote it in cm to 1 decimal place. This is because although there are numbered cm marks on my ruler, there are 10 marks between each cm. These marks are the SCALE DIVISIONS on the ruler. The smallest scale division is worth 0.1 cm here. We call it the RESOLUTION of the ruler - the smallest reading it could possibly take.

 Next job is to fill the tray with water to a depth of about 0.5 cm. I used a jug rather than a tap because I found it easier to control the pouring.

 Then tilt the tray at one end to a height of about one finger width off the ground. If you only put in 0.5 depth of water, it shouldn't spill!

Then take your finger out so that the tray drops suddenly. Notice that I did the experiment in the kitchen - I was worried about spilling! 

The water sloshes back to cover the empty bit at the raised end. See below. When it hits the end at A it makes a wave. Start timing when the water hits end A.

The wave will look like a ripple heading along. Stop timing when it hits end B. It took just over 1 second for mine.

Do A to B three times so that you can calculate a mean. Then do A to B to A. It is twice as far so we should expect it to take twice as long if it is PROPORTIONAL.

You need to keep going until you have timed the waves doing FIVE lengths of the tray if you can. It was hard to see the ripple for the fifth length because so much energy was dissipated but it was tehre if I looked  carefully. Here is the start of my results table.

When you take repeat readings, you get a RANGE of results. RANGE means (biggest repeat reading - smallest repeat reading). I have circled the three repeat readings. The biggest is 1.31 seconds and the smallest is 1.28 seconds. So here range = 1.31 - 1.28 = 0.03 seconds.
If you get asked for UNCERTAINTY for your repeat readings, UNCERTAINTY means HALF THE RANGE. Here that means 1/2 x 0.03 = 0.015 seconds.

Surveying Wigton: looking for benchmarks

This morning's Watchtree running challenge for lockdown was to find benchmarks and trig points. The history of the mapping of the UK is fascinating. Working out the height above sea level was roughly done by starting at sea level, finding the angle upwards and the distance to the next fixed point, and then calculating the height gain. The precise point for the height was chiselled into a wall with an upward arrow to empasise it. The First Geodetic Levelling was between 1840 and 1860. This benhcmark on the wall of the old Quaker chapel dates from that time.

At this point, mean sea level was measured from Liverpool. They had a second go between 1912 and 1921 and a third go between 1950 and 1968. The third go used Flush Brackets, like these on St Mary's and St Cuthbert's

St Mary's also has what is called a Bolt

We found ordinary benchmarks that probably date from the original survey at Lessonhall

 On the bridge at Waverbridge

On the milestone in Waverton

On the corner of New Street

And a couple of goes on the station bridge

There is also the trig point on the Silloth road above Waverbridge. I think this might be for mapping distances rather than heights - I'm not quite sure yet how they interact with benchmarks.

Another centre of mass experiment

Remember that the centre of mass means the single point where the mass of a spread out object seems to act.
To do this experiment, I used my protractor to make a big angle measurer. You can make your own if you like. You could choose to print it or just display it on a screen to help you measure an angle.

 You need an empty bottle WITH A LID ON. The centre of mass of all of that plastic will actually be in the centre - marked with an X on the photo. Stand the bottle in front of the angle measurer, lined up with the vertical line.

 Then lean the bottle over to the side. If the angle is very small, then the bottle won't fall over. It will wobble but then go back vertical. What I want you to write down is roughly the angle at which it won't go back vertical and actually falls over. It won't be very accurate. Try best of 3!

 I also tried the water bottle below. It has a wider base which might give better results.

 Now fill the bottle roughly to a quarter full with water. PUT THE LID BACK ON SECURELY - don't let it spill water when it tips. The water is much heavier than the plastic so now the centre of mass moves down to the middle of the water. So if the water goes 1/4 of the way up the bottle, the centre of mass is half that or 1/8 of the way up the bottle. Find out the angle of lean for which it first falls over.

 Repeat with the bottle
1. Half full.
2. Three-quarters full of water.
3. Totally full of water.
SCREW THE LID ON TIGHT EACH TIME!

Why do you add percentage uncertainties?

This is a very regularly asked question. You are multiplying two readings together and yet you ADD the percentage uncertainties. Why?
Here's how it goes. Suppose that we are calculating w by multiplying two readings called x and y. So w=xy. The absolute uncertainties are shown below.

 Since the absolute uncertainties are small, we say that delta(x).delta(y) is negligibly small and ignore it. Also, since w=xy, that cancels on both sides. Now divide what remains through by w but remember that w=xy.

 Look! The fractional uncertainty in w = fractional uncertainty in x ADD fractional uncertainty in y.

Finally multiply through by 100 and you'll see that when you multiply x and y, you ADD their percentage uncertainties.

THE SAME APPLIES FOR DIVIDING so eg if w=x/y, we still ADD the %Us.

Absolute and fractional uncertainty

When you take a reading there is always some uncertainty due to the need for a human to read off whereabouts it comes against a scale division. We write this uncertainty as a +- number after the reading and this is called the ABSOLUTE UNCERTAINTY. In advanced theory, it is written using the small Greek letter delta (squiggly d) as shown below.

 Absolute uncertainties are important if an equation involves ADDING or SUBTRACTING. Suppose we wanted to calculate the temperature change from two readings. First reading = 293+-1K and second reading is 373+-1K. So temperature change is 373-293=80K. Because we did a subtraction calculation we have to ADD THE ABSOLUTE UNCERTAINTIES. So Temperature change = 80+-2K.
You add absolute uncertainties when you subtract two readings; you also add absolute uncertainties when you add two readings.
The FRACTIONAL UNCERTAINTY is shown below

All we would need to do now is x100 to get the percentage uncertainty. So %U = fractional uncertainty x 100.
The rule for fractional uncertainties is the same as for percentage uncertainties. In any calculation where you multiply OR divide readings, you always ADD %U or fractional uncertainties.

Restarting the statistical experiment

This is where I finished my first go at electron energy level shuffling. I realised it was impossible. The idea is that there is a fixed amount of energy in the system. For an electron to go up one level, another electron must lose energy and go down a level.The accounting below didn't work - 2 up and 3 down.

 The problem was about what to do if an electron and the bottom level was told to go down. Now I have decided to roll both dice again. This is what i got after 10 rolls which looks better.

Aureole of the Sun's optical corona

Seen from Earth, the Sun subtends an angle of 0.5 degrees. A little finger at arm's length subtends 1 degree. In the picture above, as the sunlight passes through thin high cloud the light is spread. It subtends about 4 degrees. This is an optical corona rather than the halo that I saw a couple of week's ago. This time the diffraction is through water droplets rather than ice. I saw only the central bright spot up to the first minimum which is called the aureole. Full pictures can be seen here https://en.wikipedia.org/wiki/Corona_(optical_phenomenon) I need to find out if the aureole is basically the same as the Airy disc that you see when light is diffracted through an aperture and then to work out how a hole and a collection of water droplets can produce the same pattern.

Investigations in statistical physics

Thirty years ago the wonderful Nuffield A Level Physics course had this experiment that is actually a good introduction to university level statistical physics. I've never really done it so I thought lockdown would be a good time. I set up an electron energy level diagram and put 6 electrons on the first level above the ground state.

 I rolled two dice. The number on the green one sends that electron up one level and the black die sends that numbered electron down one level.

 This is what the dice roll above does.

 I did 10 more dice rolls and this is where I got to for today.

There's going to be a lot more of this!

Year 10 Finding the centre of mass

Big objects have their mass spread out all over the place which makes it hard to do calculations. Things are much easier when all the mass is concentrated in one small point. Turns out that for a big object, it is possible to fake such a tiny point. It is called the CENTRE OF MASS and it makes the calculations work. The centre of mass is the pint at which the whole mass of an object SEEMS to be.
You are going to find the centre of mass of a weird shaped object.
1. Find a bit of a cardboard box. (If you really can't find any cardboard anywhere in the house, try it with a piece of paper - preferably glue more than one piece of paper together to make it thicker).  Draw a shape onto the cardboard. Make it more complicated than a square, circle or triangle please! I choose to draw Great Britain but you don't have to be that complicated.

 2. Cut the shape out. Force a round pen through it and hang it off the edge of a table with a heavy book on the pen. Make the hole slightly bigger than the pen so that when you pull the shape to one side, it swings backwards and forwards a bit.

 3. Find a piece of string or a shoelace or something stringy. (If you REALLY can't find that, see point 4 below on what to do instead) Tie the string round a rubber or a heavy-ish object and hang it from the pen.

 Hold the cardboard and the string still with one hand and then as best you can draw a line down the string.

 4. If you really can't find anything stringy, let the shape swing freely and then hold a ruler as near to vertical as you can against the pen. Hold the ruler and the cardboard still with one hand and draw down the ruler as best you can with the other hand. This is not as good as the string method because you have to decide whether it is vertical. Gravity will pull the rubber on the string so the string really will be vertical.

 5. Do this for at least 4 different hanging points on your object. The lines should roughly go through the same point. That point is called the centre of mass.

 Notice that if you hang an object, the centre of mass will always be on a line directly below the hanging point.