Doppler Effect

Imagine a Formula 1 car approaching the stands at 60m/s. The frequency of sound made by the engine as heard by a stationary observer in the stands is higher than the actual frequency as heard by the driver. The sound is squashed up – or better, the apparent wavelength is decreased and the apparent frequency increased.

car approaching observer
car receding from observer

 

As the car recedes away from the stands, exactly the reverse happens. the observer waves goodbye to the red line. Think of EEEEYOWWWW as the car approaches then recedes.

For a stationary observer and a moving source, we can write:

These will mostly do – but IB requires us to use these as well:-A quick calculation shows how the first equation works. Let the car be moving towards us in the stands at a speed us of 60m/s and emitting a frequency f of 800Hz.

Speed of sound in air is 340m/s. We can find the frequency f’ as heard by the stationary observer. Common sense tells us whether we add or subtract the velocities – in this case, we subtract and hear a higher frequency as it approaches us. (the EEEE bit)

 

As it recedes, we add, thus: (the YOWWW bit)

Police speed detectors bounce microwave radiation (about 10GHz) off a moving vehicle and detect the reflected waves. Because the car is moving towards the police observer, these waves are shifted in frequency by the Doppler effect and the difference in frequency between the transmitted and reflected waves provides a measure of the vehicle’s speed. Of course it works just as well for recession speeds as well.

Two Doppler shifts because of the reflection from a moving target. c is of course the speed of light

By observing distant galaxies, Edwin Hubble concluded that distance and recession speed were proportional – so galaxies further away are receding faster than closer galaxies. We know this because the atomic fingerprint or spectrum of atomic hydrogen or helium is shifted to the red (long wavelength) end of the visible spectrum. The degree of redshift can be used to find out how far away a galaxy is.

This absorption spectrum shot (idealised) shows what the spectrum of atomic hydrogen might look like from several distant objects like galaxies. The further away, the greater the redshift. Redshifts of up to 0.95c have been observed – the light having taken almost the lifetime of the Universe to reach us.

 

 

Finally, a medical use. Doppler blood flow is a technique whereby ultrasound waves (f about 800Hz) emitted from a piezoelectric transducer (transmitter/receiver) are reflected off red blood cells in an artery or vein as they are moving towards the stationary detector. The more occluded or blocked the artery is (think about a fluid in a pipe) the faster the cells are moving. It can also be used to find blood clots in deep veins – DVT – deep vein thrombosis – can be fatal.

The detector and the moving cells are at an angle hence the cosine term and, like the police car, the factor 2 accounts for the reflection from a moving source.

 

Olber’s Paradox.

The comedian George Carlin used to do a sketch where he was a kind of hippy weather man. “And the weather tonight is … dark, man.”

But he didn’t ask the question, why is it dark? Once you get past the flippant answer “duh, because it’s night-time”, you see why it is an interesting question. If there are an infinite number of stars, shouldn’t the sky be bright?”

Since the sky isn’t very bright at night – a lot of dark plus a few pinpoints of light – this is called Olber’s Paradox. Put simply, if the universe is infinite then wherever you look you should see so many stars that the night would be brighter than the day.

Johannes Kepler considered this question but he argued that the universe must be finite. Otherwise the total flux from all the stars would make the night sky “as luminous as the sun.”

Suppose we gaze out in any direction from picture1Earth, imagining a thin sphere of radius R around us. Unlike Kepler, Newton’s model assumed a uniform, infinite (and static, or not expanding) universe, the number of stars in the shell is proportional to R2 and the intensity of radiation from the shell reaching Earth is proportional to 1/R2. So, according to Newton’s model such shells stretch to infinity so the sky can never be dark.

Of course, now we know that the Universe is expanding – it has a beginning – and stars and galaxies aren’t tastefully arranged in neat little spheres.

Lots of bits are dark and the combined effect is to make the night sky dark.

Furthermore, there’s more ‘dark’ than ‘light’ and the Universe expansion probably isn’t uniform. Best estimates suggest that if we calculate the energy needed to overcome gravity, dark energy ( the stuff that accelerates the expansion of the Universe) makes up roughly 68% of it. Dark matter makes up another 27%, leaving the “normal” or baryonic matter that we are familiar with to make up less than 5% of the cosmos.

 

Very Basic Thermodynamics

Screen Shot 9.pngThermodynamics is the study of energy. (IB Core: Section 3.2, Option B.2 part)

A MOLE is an amount of stuff ~6×1023 particles’ worth. (SI unit)

This number is the Avogadro constant, NA (mol-1) – the number of constituent particles in 1 mole of substance. In 12g of C12 or 18g of water, there are 6×1023 carbon atoms or water molecules respectively – one mole. One mole of electrons contains 6×1023 electrons, and so on.

We should remember that for a fixed mass of an ideal gas, the ideal gas equation (below) applies The equation is considered most accurate for monatomic gases at high temperatures and low pressures. Check the link so you understand the assumptions of the kinetic theory of gases.

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so k=R/NA

Temperature is a measure of the degree of hotness of a body, as compared to a fixed scale. Normally we calculate in kelvins (K) – a base unit – where a difference of 1K corresponds to a difference of 10C. For now, from the Ideal Gas Laws,Screen Shot 10.png

Energy exists in many forms, such as heat, light, chemical energy, and electrical energy. Energy is the ability to bring about change or to do work.

Laws of Thermodynamics                               Zeroth Law

If A is in thermal equilibrium with B and B is in thermal equilibrium with C then A and C are also in thermal equilibrium.

All thermal equilibrium means is that the rate of transfer of heat from A to B is the same as that from B to A. If B is hotter than A, B transfers heat more rapidly to A than A does to B. But – the transfer is still two-way. Temperature difference between two bodies determines the net flow rate of energy between them.

First Law

The First Law of Thermodynamics or Law of Conservation states that the total energy in the universe is always conserved; it cannot be created or destroyed.  Energy can only be converted from one form into another. For a fixed mass of an ideal gas, the gas can either do work on its surroundings, delta W, gain heat from its surroundings, delta Q or its internal energy increases. delta U change in internal energy is a function of temperature: U is a large scale concept. We cannot talk about the “thermal energy” of something – it has no meaning. Instead we refer to the internal energy of a body which is the total potential energy (arising from intermolecular forces) + random kinetic energy (translational and rotational) of the molecules in a sample of material and clearly we can only measure change in (delta) U not U directly. Easier to see with symbols.Screen Shot 1.png

Let’s imagine a frictionless piston, containing a fixed mass of an ideal gas (very low density, pressure and high temperature). Now, let’s make some changes to it. Isothermal changes are changes without change in temperature, thus the internal energy of the gas is unchanged. So, all the heat supplied = all the work done by the gas. Take note of which processes are slow (so heat gets in or out) and fast (no heat in or out)

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Adiabatic Process. No heat in or out. Any work done is done fast on or by the gas and is reflected in a change in internal energy. Pure adiabatics are quite rare.

Screen Shot 3.pngThe nearest thing to a fast adiabatic process is a bursting tyre. The rubber of the tyre is an insulator, so no heat enters or leaves the gas and the work done by the gas on escaping through the hole is at the expense of a fall in temperature of the remaining gas inside the tyre. On the graph below, this would be the thick green line from high T to low T

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NB: the AREA UNDER the pV graph is the work done on or by the interaction. (fave exam question where you have to count squares)

You might now ask yourselves what the pV graph might look like if the volume was NOT allowed to change – the “sticky piston” problem called an ISOVOLUMETRIC change

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and, yes, it’d be a vertical line at some constant volume between two isothermals.

An ISOBARIC (constant pressure) line would be a horizontal line between two isothermals. It might be helpful to sketch both on the graph above.

A CYCLE of events means that we make changes to the gas to get back to our starting point. The (shaded) area swept out by these changes is a measure of the work done during one cycle. T1 is greater than T2, clearly. Here’s an idealised diagram. In reality heat enters (BC) and leaves  DA) the gas so BC and DA aren’t perfect adiabatics in real systems.Screen Shot.png

This looks complicated, but it’s not, really. Please make sure you look at the problem below from IB 2008. As you can see it consists of two isobaric and two isovolumetric events.screen-shot

Now calculate the overall energy transferred in one cycle and explain whether, after one complete cycle, the internal energy of the gas goes up, down or stays the same.

One-Way Processes.

The Second law of thermodynamics states that that the entropy or measure of disorder of an isolated system always increases, because isolated systems spontaneously evolve towards thermodynamic equilibrium – the state of maximum entropy (or minimum potential energy – a ball spontaneously rolls down a hill and not vice-versa.) A cup of tea tends to cool as energy is dissipated to the surroundings and not vice-versa. An increase in entropy is the Universe doing the most likely thing – the probable is what usually happens. (when you blow up a building it tends not to spontaneously reassemble as if the film ran backwards)

A mechanical watch will run until the potential energy in the spring is converted, and not again until energy is reapplied to the spring to rewind it. A car that has run out of gas will not run again until you refuel the car. In the process of energy transfer, some energy will dissipate as heat. Entropy always increases and is a measure of the disorder of the Universe – put another way, the more energy is transferred from one body to another the greater are the number of ways in which that energy dissipation can take place. For example, a waterfall turns a paddle wheel which drives a turbine which turns an alternator which produces electricity, dissipating energy into many different forms along the way.

IB: Engineering Science Option B: Torque, Angular Momentum and Moment of Inertia (amended)

THIS POST  WAS ORIGINALLY WRITTEN FOR MY OWN IB CLASS. THERE ARE HANDOUTS AND PROBLEMS HERE THAT WE DID IN CLASS BUT NEWCOMERS SHOULD FIND THEM HELPFUL. GO AHEAD AND TRY.

It’s useful to bear in mind that if you can do SUVAT problems, you should have no trouble with their circular equivalents.

Four handouts in total to download; please make sure you work through them carefully. Any difficulty, get in touch.

The main arguments here are the idea of rotational motion and torque as force x distance from pivot x sin(angle between them)

Moments of Inertia need not be calculated for this course – if necessary, you’ll be given them. However, here’s a little problem to think about. You have 2 balls of identical diameter and weight. One is solid, one is hollow. You can’t tell which is which just by knocking on them. Devise a simple way of finding out which one is the solid one (hint: think about the balls rolling down an inclined plane from the same height. Now, compare the moments of inertia of the two balls. The rest is conservation of energy so quite easy.) If you can’t write out the solution, message me for help.

This little animation is quite fun to glance at – the angular displacement is, however, in degrees, not radians, so be careful.  Here’s a screenshot, showing displacement – time graphs for the ant and the ladybug – the constant time period implies constant angular velocity. Notice too that the ladybug leads the ant by 90 degrees

Screen ShotNotice, angular velocity is constant, but the linear speed of the ant and the ladybug are not the same. The ladybug, being closer to the axis of rotation has a smaller linear speed, because:screen-shot-2016-12-07-at-12-20-20(Notice the vertical displacements of the bugs execute SHM – AHLs will study this later)

  1. A review of circular-motion-1
  2. Key Ideas, torque and couple
  3. Basic concepts about moment-of-inertia
  4. From the Specimen Paper specimen-question-for-option-b

Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis. Just as inertial mass is the ratio of linear momentum to speed – its resistance to acceleration, in other words:screen-shot-2016-11-29-at-17-57-19 Angular momentum in a closed system is, just like its linear counterpart, conserved. The ice skater rotates faster when the arms drop to the sides because moment of inertia is reduced and thus angular velocity increases.

Look at the Wolfram demonstration Screen Shot

You will need to download the Wolfram CDF player in order to run the demo.

We might also notice that, for a body starting to rotate from rest:screen-shot-2017-01-14-at-7-45-20-pm

Practically, we find I by imagining a flat sheet of any shape like this having an infinite number of mass elements m at their respective distances r from the pivot, each contributing  torque about the axis of rotation.

We have to add all the torques up, normally requiring integration. But, to keep it simple, we can write:

screen-shot-2016-11-30-at-18-24-25

which is, as the handout shows, the basis for finding I for lots of other shapes and axes of rotation. Remember, you’ll be given I for a particular shape as required.

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Bear in mind that when we do problems, the total energy of the system is the sum of the rotational and linear parts – important when we think about an object rolling (instead of sliding) down a hill, for example. Take a look at this solid-cylinder-rolling-down-an-inclined-plane. which runs through a few basic ideas plus some possible lab work.

Finally, for now, a use for all that stored energy.

screen-shot-2016-11-16-at-15-49-54

The great flywheel on Richard Trevithick’s 1802 locomotive, used to level out the power supplied by a single cylinder. Rotational inertia kept the wheel turning.

Energy Transformations (2) EPE to KE: The Battle of Agincourt

A bow stores elastic potential energy in the flexible ash of the wood of the bow and the stretched string.

Screen Shot 2016-06-27 at 14.11.32

The English archers won the Battle of Agincourt in 1415 because the range of the better-made and longer English bows was greater than the bows used by their French enemies. They stored and released energy more efficiently.

Screen Shot 2016-06-28 at 18.01.11They could probably count on a release speed of 100 feet per second = 3000 cm/s = 30m/s and they knew that a 45 degree angle gave maximum range.

 

So 30 cos 450 = horizontal speed = 21.2m/s. This stays the same because  a=0 in the horizontal plane.

In the vertical plane, we have to use the equations of motion (constant a). With 30sin450 as u, and acceleration = g (9.81ms-2)  this  yields t = 2.16s going up, PLUS the same coming down so the arrow’s total time in the air = 4.32s.

So, range =21.2×4.32 = 91.6m…and vertical height reached is  33.2m as long as air resistance is neglected.

Now, let’s take this apart…

At the top of its flight, what kind of energy does the arrow have? Estimate arrow mass at 0.1kg (is this reasonable?)

A:  Using kinetic energy in the horizontal direction (22.47J) plus GPE at maximum height (32.57)

Estimated energy 55J

Where does this energy come from? Think about the area under a F/x graph for a spring or similar where Hooke’s Law is obeyed… Screen Shot 2016-06-27 at 13.12.08.png

A: the energy stored in the stretched bowstring. Let’s assume it obeys Hooke’s Law.

Using our calculations so far, we can now estimate the maximum pulling force the archer would have to use. Suppose the bow is drawn a distance of 0.5m

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With quite a lot of ifs and buts – you can see where the approximations are – the archer would have to pull with a maximum force of 55/0.5N or 110N. This would be like holding a mass of 11kg just off the ground. With one hand…

 

Heisenberg’s Uncertainty Principle. Confining a wave/particle in a box.

In subatomic terms, because of wave particle duality, certain pairs of measurements such as where a particle is (x) and  where it is going (its position and momentum) cannot be precisely known. If we know one very precisely, the other cannot be known. Putting this another way, a particle has mass (hence momentum) also a wavelength given by the de Broglie expression

Screen Shot 2016-06-03 at 09.29.37

When particles’ wavelengths interfere, they form a wave packet of finite size having a length which has to fit into the confining box, which can happen in a variety of ways…Here, we’re only really concerned with the smallest “wavefunction”, shown in red at the bottom. The diameter of the box is approximately half a wavelength. The rest are there just to show what’s possible. A wavefunction represents the probability of finding the wave in a particular space.

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Confining our wave in a box, where it is and its momentum are defined like this:

Screen Shot 2016-06-02 at 16.39.04

This makes sense in the context of a problem. Imagine an alpha particle confined within a nucleus of gold. Given the alpha particle has a wavelength confined by a ‘box’ the size of the nucleus, whose diameter might be:

Screen Shot 2016-06-02 at 16.48.20

Suppose we want to find the energy of the confined alpha particle. We use:

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The energy can be found using a different expression:

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We can find the mass of an alpha particle (2 protons and 2 neutrons. If we plug in the numbers, we get 4.3×10-15 J or 27keV, consistent with observed energies.

You might try the same calculation to find out the energy an electron would have to have to confine it inside the nucleus.

This is why we don’t get electrons inside nuclei…