Resonance

A long string, under tension from a weight at one end, is mounted horizontally; the other end connected to a variable frequency mechanical oscillator (right). A transverse standing wave can be set up. This is a well-known class demonstration.

As the frequency is increased, at certain frequencies, a series of large amplitude nodes and loops appears. This is an example of resonance. For more detail, look here

The resonant frequencies are determined by the mass per unit length of the oscillator (bass strings have a natural frequency lower than treble strings), also the tension in the string, thus:

Resonance occurs when a driven oscillator is made to vibrate at the same frequency as a driver oscillator. Resonance effects are characterised by vibrations with large amplitude.

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 simple harmonic  oscillators are nothing more than periodic exchangers of KE and PE.

Here is a link to  Barton’s Pendulums. You will see this in class. In fact, you might make a set yourselves.

Here’s a video produced by a student

Also – see if you can figure out what’s going on here….

Think about a child being pushed on a swing.

The pusher (driver) pushes at the same frequency as the natural frequency of oscillation of the swing (driven). The result is that the swing oscillates with large amplitude, since the energy it needs to gain amplitude is being supplied at exactly the right time in the cycle. The pusher and the pushed are a quarter of a time period out of phase.

These resonance effects are common. The bus side vibrates as it pulls away, since the metal skin of the bus has the same natural frequency as the engine at low revs. Imperfectly balanced washing machines ‘walk’ across the floor during the spin cycle since the frequency of the vertical component is the same as the natural frequency of vibration of the retaining springs, causing vibrations with large amplitude.  Sometimes , these can be disastrous.  Check out the video of the famous collapsing bridge over Tacoma Narrows in Washington State, called “Galloping Gertie”. Amazing! The bridge collapsed in November 1940 because of a resonant effect between the material in its construction and the wind whistling down the gorge, rather like blowing a reed wind instrument to make a loud sound.

Chladni figures are complex resonant vibrations – or two-dimensional standing waves –  made on a flat sheet and this Irish teacher’s take is very good indeed. Originally, Robert Hooke (yes, him again!) experimented with drawing a violin bow across flour-covered glass plates.

Ernst Chladni  (photo) used the same trick with metal sheets, subsequently with a magnetic oscillator in order to fine-tune the frequencies more accurately. They’re quite fun to watch.

Expensive guitar tables have complex Chladni patterns which account for the characteristics of the instrument.

Chladni figures on a classical guitar

Mass-Spring Oscillator (updated)

Read the IG post on Hooke’s law before proceeding. Check out a Mass Spring Oscillator as a moving .GIF.

In order for mechanical oscillation of a mass/spring system to occur, it must possess two quantities: elasticity and inertia. When the system is displaced from its equilibrium position, elasticity provides a restoring force such that the system tries to return to equilibrium. The inertia property – resistance to changes in motion – causes the system to overshoot equilibrium. This constant interchange between elastic and inertial properties is what allows oscillatory motion to occur. The natural frequency of the oscillation is related to both the elastic and inertia properties. Weak, springy springs (small elastic restoring force) with large weights (large inertia) on them have a long time period and vice-versa.

The weight of the added mass acts vertically downwards, the restoring force on the spring acts upwards.

The elastic constant of the spring is k.

At any other displacement, the acceleration = a

When the mass is displaced downwards the restoring force acting on it, F, is of magnitude kx, upwards. (kx is larger than weight)

If it is displaced upwards the net force (and acceleration) is downwards (kx is smaller than weight)

So:

A graph of T² against m is linear through the origin with slope = 4π²/k

Music of the Spheres

Musical appreciation has much to do with agreeable resonant frequencies.

Pythagoras discovered relationships between musical notes. The pitch of a note being played on, say, a guitar depends on:

• The length of the string.
• The tension of the string.
• The material the string is made of.

Pressing a finger on the string clamps the string on to the metal fret to the left of where the finger is pressed. The effective length of the string becomes the distance between the fret and the bridge. If we say the length of the string is one unit, we can show the effective lengths of the string for each of the notes in the scale as fractions of the whole, related by powers of 2 and 3. These harmonious “consonant” lengths were discovered by Pythagoras. Furthermore, Pythagoreans believed that this was a general principle of the universe: everything is related by the ratios of whole integers..

Soap Chemistry

 

It is hard to say when soap was first invented. Some suppose that even prehistoric man (or, more probably, woman) had a primitive form of it at their disposal. Whether the hunters and gatherers had soap is debatable, but it is certain that soap was available to the ancient Babylonians. Soaps have been excavated in clay cylinders that date back to 2800 BCE. By 1500 BCE. Egyptian medical scrolls recommend a soap made from alkaline salts and animal and vegetable oils for skin conditions.

Later, the ancient Romans discovered the cleaning power of soap accidentally. At Mount Sapo, where animals were sacrificed, rain mixed animal fats, wood ashes and clay into the soil. Incidentally, women washing their clothes by the stream found it was much easier to wash their clothes with some of this clay mixture. Legend links Mount Sapo with the process of soap making (saponification). Interestingly, although Romans are famous for their baths, they actually did not use soap to wash. They coated themselves in oils and then used a scraping tool called a strigil to clean their bodies. However, bars of soap were found in the ruins of Pompeii and archaeologists believe soap was used for laundry and occasionally on the body.

Now for the chemistry. Check out this link. The whole nine yards…

Catalysts

…increase the rate of a reaction while remaining unchanged chemically themselves.

There are two distinct processes…

1. The reactants are ADSORBED on to the catalyst surface, making it easier for the reactants to combine. The reactants like the surface of the catalyst, so they tend to move towards it, making collisions and hence reactions more probable.

2. The catalyst undergoes an intermediate, short-lived reaction, returning to its original state after the reactants have reacted together. Biological catalysts, called ENZYMES work like this.

For more details, download this  Word document explaining how catalysts work

Alcohols

A surprisingly large number of common foodstuffs are produced by direct fermentation of yeast or similar microbes, or by using the principles of fermentation where sugars are digested by micro-organisms to form alcohols.

 

 

 

 

 

 

 

 

 

 

Click alcohols to understand basic alcohol chemistry.

Linear Air Track: Measuring Acceleration (for IG and AS)

The trolley floats on a cushion of air on a V-shaped track, so there’s no friction to slow it down. When the light beam is interrupted by the passage of the trolley it starts a clock. The clock stops when the beam is restored. The logger can be connected to a computer for easier display and configured to calculate average speed.

It's not working...Yes, it is.

IG: The card on top of the trolley measured 199mm = the distance travelled. The time for the trolley’s journey through the light gates can be read off the display.

Example:

Average speed = distance / time = 199mm/0.403s = 493.8mm/s (1d.p)

AS students. You MUST read this part: If one end of the trolley was inclined (tipped up), the trolley will accelerate down the slope since the component of its weight acting parallel to the slope will provide the unbalanced force. Can we design and carry out an experiment to compare the acceleration of the trolley with the angle of inclination which will determine the size of the unbalanced force?

I want you to develop your own procedure for this and try it out in the lab, thinking through each stage carefully.

The new light gates are more sensitive and better than the old ones.

If we use two light gates the logger can calculate the average speed of the trolley as it passes through the first and then the second light gate. The onboard timer will then be used to calculate the acceleration of the trolley being the difference in average speed between gate 1 and gate 2.

Perhaps a better way might be to use just one light gate with a split card, like the one shown below.

The lengths of the two uprights and the space between them must be measured. The logger uses (v-u)/t to calculate acceleration.

If we input the dimensions of the card into the logger, it will calculate the average speed during the first and then the second pass of the card and calculate the acceleration for us automatically. If you can’t remember how to do this, look here.

Important design points:

1. How are you going to ensure that the air track is horizontal to start with? Should each run start from the same place? How will you make sure that this happens?

2. You can use plywood spacers or even new exercise books to prop up one end of the track. How are you going to measure the angle to the horizontal? (not with a protractor)

3. What about accuracy? How many times should each measurement be repeated and what are you going to do with the values?  Can you identify any errors in your procedure or results? What are you going to do about them?

4. How are you going to show what your results look like – a table, a graph, or both?

5. Can you predict what the shape of the graph ought to be? What information is obtainable from it?

Solution. Check carefully..All of you should have found time to use the apparatus. AS asks about datalogging principles.

An exaggerated view. The free body force diagram plus resolved components is shown. Weight W acts vertically downwards and normal reaction acts R at 90 degrees to the surface. The resolved component of W, W sin ϑ causes the trolley to accelerate down the slope. F(R) is a zero frictional resistance so can be ignored.

Keep It Simple…

Keep it simple…

When we do science, we tend to over-complicate things, so we need to remember to keep it as simple as we can. Occam’s razor is a logical principle attributed to the mediaeval philosopher William of Occam and it states that ‘one should not make more assumptions than the minimum needed’. Put another way, ‘the simplest explanation is usually correct’.  If you like it in Latin – we can write it as “lex parsimoniae“. So, if we have two competing hypotheses that make the same predictions, we should select the solution with the fewest assumptions. It is not meant to be a way of choosing between hypotheses that make different predictions.

It underlies all scientific modelling and theory building. I call it “the tight-wad principle” – never spend any more energy, money or effort than you need to solve the problem. OK? That’s settled then.  As and when I get time, I’ll post some silly stuff here to entertain, elevate and amuse. Watch this space.

Simple Pendulum and SHM

Galileo’s Lamp (Pisa Cathedral)

Galileo showed that the motion of a swinging pendulum is ISOCHRONOUS – amplitude is independent of time period T.

Without derivation, it can be shown that

for a simple pendulum

The rules for measuring oscillations are as follows:

• rigid support
• heavy bob
• massless string
• measure l to the centre of the weight
• small oscillations (less than 5 degrees either side – or amplitude less than l/6)
• count 10 or more oscillations from the CENTRE or 0 of displacement, seen against a vertical fiducial mark, 0,1,2,3….and so on, divide by 10 to get T.
• repeat as time permits.

You are penalised for not drawing a labelled table of results including headers, units, repeats and treatment of errors if appropriate.

In the absence of damping forces tending to decrease amplitude and increase time period, for small oscillations (angle of swing less than 5 degrees) the oscillation is simple harmonic.

You will do an experiment in the lab to show that T²  and l are directly proportional. Here are some specimen results.

Calculate the slope of the graph. Use it to work out a value for g since gradient =  4 π²/g

This applet contains  more than we need, but it’s worth a  look. You can change l, m and g.

Knowing that SHM is executed and by definition:

at the extremities of the oscillation, the force toward the centre is maximum, hence a is maximum and proportional to x.

knowing that

Conical Pendulum

This little applet shows how a conical pendulum works . Check it out.

Make sure you can derive the equation 

tan θ   = v²/rg

The Periodic Table

IF you think this is hard, blame Dmitri Mendeleev, who presented a paper to the Russian Chemical Society in 1869, illustrating recurring or “periodic” trends in the properties of the elements. There were a few gaps, which got sorted out in the following years.

Great Uncle Dmitri

Did you know…you can actually buy a shower curtain with the periodic table of the elements on it?  This a great site from Los Alamos Laboratory (where the atomic bomb was developed in the 1940’s) and tells you almost everything you might need to know.  First, a few navigation rules.  The icons at the top show you what to do and where to go and clickable links to each element give basic information. They even tell you that there’s an element (118) which might not even exist!

To Year 8…I’m sorry. There are only 31 flavours of Baskin Robbins ice cream. I was certain there were 33!

Simple Harmonic Motion – SHM for a mass/spring oscillator

We start with a statement of Hooke’s Law, F=kΔx, straight line through the origin. We then displace the spring a further value x, release and let go, watching the spring oscillate.

A few curves showing how displacement x, velocity v and acceleration a vary with time. We see that when x=0, a=0 but in the opposite direction also when x=0, v=maximum. Put another way, when v=0 both a and x are maximum. It helps to study these curves carefully.

How did we get them? The vertical component of a rotating vector – imagine a mass moving at a constant speed in a circle – executes SHM. Here’s an animation. A sine curve or a cosine curve is produced, depending on where you start.

Plotting the displacement, velocity and acceleration on the same axes looks like this.

Dynamics of SH oscillator

Print out a copy of the graph and measure the slopes of the x/t and v/t graphs to find instantaneous velocities and accelerations. (this isn’t a proper exam method, btw so check your answers by calculation. Exams sometimes penalise if you try to measure slopes unless specifically asked to do so.)

The graph shows that for a body moving with SHM, displacement is proportional to acceleration, but opposite in sign. In an exam, it’s better to say that a is proportional to x and always directed to the zero of displacement. Starting from x=0 we get a sine curve.

Since max v = rω, velocity varies with time as:

The slope of the v/t graph gives the acceleration, so:

but x = rsinωt, so

The constant of proportionality is ω² 

f being the frequency of oscillation (Hz) = 1/time period, T

This equation defines motion which is simple harmonic.

Use the graph  to find the time period of the oscillations.

Answer: (careful – vertical and horizontal axes must match – they do here…)

Diffusion

Before we start, when sugar is dissolved in water, the sugar molecules diffuse in the gaps between the water molecules, which we understand as dissolving.

You might like to look at this animation. It illustrates what happens when a perfume bottle or room freshener is opened.

IG students look a little more deeply. To show that larger molecules diffuse more slowly than smaller ones, we use a long horizontal glass tube with stoppers fitted at both ends. Using tongs we dip one piece of cotton wool into concentrated hydrochloric acid and another piece into concentrated ammonia solution. This is nasty stuff, so we use a fume cupboard. Drain off excess liquid. Simultaneously, we put the soaked pieces of cotton wool inside the ends of the glass tube. Close the ends of the glass tube with the stoppers. Watch for a white ring forming where the ammonia gas and the hydrogen chloride gas meet after diffusing through the air towards each other.

Ammonia molecules are less massive therefore faster moving than heavier hydrogen chloride so the white ring of ammonium chloride should form nearer to the hydrogen chloride end of the glass tube.

The reaction is as follows:

hydrogen chloride    +    ammonia   <=>     ammonium chloride.
HCl_(g) +     NH_3(g) <=>           NH₄Cl_(s)

(The reaction is reversible, so the arrows go both ways)

Alkenes, alcohols, carboxylic acids

Alkenes are a homologous series of unsaturated hydrocarbons – this means that their C atoms contain double bonds. No. of Carbons Root Name Formula CnH2n Structure 2 ethene C2H4 CH2=CH2 3 propene C3H6 CH2=CHCH3 4 1-butene C4H8 CH2=CHCH2CH3 5 1-pentene C5H10 CH2=CHCH2CH2CH3 Alkenes are also called OLEFINS because they form oily liquids on reaction with chlorine gas.

Ethene is the number one organic chemical synthesized in the US and the world. The small quantities of ethane, propane, and butane found in natural gas are converted into ethene. It can be produced by thermal cracking  (heating at high pressure -7 atmospheres and high temperatures  – 800 degrees C- of ethane to produce ethene and a hydrogen molecule. Alkenes are the raw materials for a number of plastics such as polyethylene, PVC, polypropylene, and polystyrene. See later.

Alcohols are compounds where a H atom is replaced by an OH group. They boil in general at higher temperatures to their corresponding alkane

Draw these out and name them.

CH₃OH           CH₃CH₂OH           CH₃CH₂CH₂OH

Carboxylic Acids, finally..

The parent chain must include the carboxyl carbon, (COOH) which, incidentally, is given position number 1. The name of the alkane attached is changed by replacing the -e with -oic acid.  They have  higher boiling points than the corresponding alkane.

You should be able to draw this one out CH₃COOH – ethanoic acid (aka acetic acid, a weak acid and constituent of vinegar)

Rates of Reaction

In a chemical reaction, old bonds are broken and new bonds are formed

If the energy released in bond making is greater than that needed for bond breaking, the reaction is EXOTHERMIC. If not, it’s ENDOTHERMIC

 

H₂ + Cl₂ = 2HCl

is highly exothermic and produces a lot of heat.

 

• Fast reaction: silver nitrate + sodium chloride = sodium nitrate + silver chloride (a white precipitate forms IMMEDIATELY the two liquids meet)
• Slower reaction: concrete sets in a matter of hours
• Very slow reaction: iron rusts (oxidises) in the presence of oxygen(air) and water

 

Concentration: If there is more of a substance in a fixed volume, there is a greater chance that molecules will collide and speed up the rate of the reaction. If there is less of something, there will be fewer collisions and the reaction will probably happen at a slower speed. 

 

Surface area. If marble chips are reacted with hydrochloric acid the following happens:

Calcium Carbonate + Hydrochloric acid = Calcium Chloride + Water + Carbon dioxide

A gas can be seen to bubble from the marble surface. If the chip size is small there is more reactant area available to the acid, so the reaction goes faster. With larger chips, the reaction proceeds more slowly and the gas is formed less rapidly.

Temperature: When you raise the temperature of a system, the molecules bounce around a lot more (because they have more energy). When they bounce around more, they are more likely to collide. That fact means they are also more likely to combine. When you lower the temperature, the molecules are moving on average more slowly and collide less. That temperature drop lowers the rate of the reaction. 

Pressure: Pressure affects the rate of reaction, especially when you look at gases. When you increase the pressure, the molecules have less space in which they can move. That greater concentration of molecules increases the number of collisions. When you decrease the pressure, molecules don’t hit each other as often. The lower pressure decreases the rate of reaction. 

 

 

 

 

Isomers and Alkane Reactions

Sometimes, questions in IG involve the naming of organic isomers. The first section of this file looks at this for branched alkanes.

The numbers used to indicate where the pendent (or hanging) groups are must be as low as possible, so remember to check from both ends of the molecule.

This is a useful resource for working out how to name isomers. Look at it carefully and follow this link to the IUPAC Names tab which is at the top of the page.

The second section looks briefly at some reactions of alkanes. This link is a particularly interesting one about the effects of methane

SLaG

 

SLaG – Solids, Liquids and Gases

SLaG-4-beginners is a Word 2003 file, so whichever version you’ve got should be fine. Download it and print it for your notes. All you need to know.

Have a look, then re-read the Atomic Traffic post

Further Travel Graphs

Straight line distance/time graphs

these tell us that the object is either at rest (the graph is horizontal) or moving at a steady speed either away from its starting point (+ or SW to NE slope) or returning to it (- or NW to SE slope)

Straight line velocity/time graphs

Remember, velocity – a VECTOR quantity since it has both size and direction – just means ‘speed in a particular direction’

these tell us that the object is either moving at a steady speed  (the graph is horizontal) or accelerating from low to higher speed (+ or SW to NE slope) or decelerating (more properly negative acceleration) from a higher to a lower speed over time (- or NW to SE slope)

What about curved graphs?

A distance/time graph curving upwards means that the object’s speed is increasing with time, and vice-versa.

For IG: recognise the shape of the graph only. For AS, you should be able to solve the problem

Travel Graphs

 

Crossing the Road

 

 

Read what follows.

Then come back and try this question. Click on the graph to make it bigger. There’ll almost certainly be something like this in IG.

When speed is constant (in other words the speed doesn’t change) we find it as shown:

Speed = distance travelled
                   time taken

so: time taken = distance/speed and distance = speed x time

Remember, when using any formula, the units must all be consistent. For example speed could be measured in m/s, distance in metres and time in seconds.

If speed does change, the average (mean) speed can be calculated:

Average speed = total distance travelled
                               total time taken

Example:
If a car travels at a speed of 10m/s for 3 minutes, how far will it travel?
Firstly, change the 3 minutes into 180 seconds, so that the units are consistent. Now rearrange the first equation to get distance = speed × time.
Therefore distance travelled = 10 × 180 = 1800m = 1.8km

Units
In calculations, units must be consistent, so if the units in the question are not all the same (e.g. m/s, m and s or km/h, km and h), change the units before starting, as above.

The following is an example of how to change the units:

Example:
Change 30km/h into m/s.

30km/h = 30/60 km/min              (1)

= 30/3600 km/s = 1/120 km/s     (2)

= 1000/120 m/s = 8.33 m/s        (3)

In line (1), we divide by 60 because there are 60 minutes in an hour. Often people have problems working out whether they need to divide or multiply by a certain number to change the units. If you think about it, in 1 minute, the object is going to travel less distance than in an hour. So we divide by 60, not multiply to get a smaller number.

Next: the reverse process

Change 20m/s into km/h

20m/s = 20x60x60m/h

Divide by 1000 to convert the m into km:

20m/s = 72km/h

Velocity and Acceleration

Velocity is the speed of a particle and its direction of motion (therefore velocity is a vector quantity, whereas speed is a scalar quantity).
When the velocity (speed) of a moving object is increasing we say that the object is accelerating. If the velocity decreases it is said to be decelerating. Acceleration is therefore the rate of change of velocity (change in velocity / time for change) and is measured in m/s².

Example:
A car starts from rest and within 5 seconds is travelling at 20m/s. What is its acceleration?

Acceleration = change in velocity / time for change = 20/5 = 4m/s²

 

Distance-time graphs:
These have the distance from a certain point on the vertical axis and the time on the horizontal axis. The velocity can be calculated by finding the slope or gradient of the graph. 

The slope = RISE (how far up) / RUN (how far along)

 

Velocity-time graphs/ speed-time graphs:
A velocity-time graph has the velocity or speed of an object on the vertical axis and time on the horizontal axis. The distance travelled can be calculated by finding the area under a velocity-time graph. Acceleration is the gradient or slope of a velocity-time graph.

On travel graphs, time always goes on the horizontal axis (because it is the independent variable).

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Vectors have size and direction, scalars have size only

Large Hadron Collider *Update*

The spectacular, and incredibly expensive LHC went on-line this week. Two beams of  ‘hadrons’ – either protons or lead ions – will travel very fast in opposite directions inside the huge circular accelerator, gaining energy with every lap. Physicists will use the LHC to recreate the conditions just after the Big Bang by colliding the two beams head-on at very high energy. Teams of physicists from around the world will analyse the exotic and short-lived particles, possibly with interim, short-lived microscopic black holes created in the collisions. Before we all think that the end is nigh, the energy of the colliding particles is less than a mosquito in flight, so the end of the world doesn’t look as if it’s going to happen because of the LHC just yet…

There are many theories as to what will result from these collisions, but it does seem likely that a brave new world of physics will emerge from the new accelerator, as knowledge in particle physics goes on to describe the workings of the Universe. For decades, the Standard Model, unifying electromagnetism, the weak and strong nuclear force, but not gravity, has served physicists well as a means of understanding fundamental physical law, but it does not tell the whole story.

Only experimental data using the higher energies reached by the LHC can push knowledge forward, challenging those who seek confirmation of established knowledge, and those who dare to dream beyond.

Shame, really. They get their hands on the most expensive machine in the world, then a fortnight later, it springs a leak, they break it and it’s gonna take two months to fix…

UPDATE

It actually took 14 months to repair a spliced copper cable which caused an explosion.

The European Organisation for Nuclear Research, Cern, said in a statement on Friday that particle beams are once again circulating in the LHC, and that a clockwise circulating beam was established at 10 PM local time.

According to the Cern Twitter feed, an anticlockwise beam was also successfully injected, and both beams have completed many thousands of turns of the LHC.

“The LHC is up and running regularly. Operators are adjusting and testing obedient beams,”

Part of the long accelerator tunnel at CERN