Friction.....Physics in the Real World

In many textbook problems we see the phrase "neglect air resistance" or "assume that all surfaces are smooth" in  projectile problems or mechanics problems. These assumptions allow problems to be "solvable"  using the well known equations of uniformly accelerated motion. Now let us step into the real world where dissipative agents that remove energy from a system act. What causes the frictional force between two surfaces? How can we explain friction at an atomic level? And, why is it "harder" to push start a heavy object than to keep pushing it along?  Finally, here is a discussion question on air friction. Imagine that a ball is thrown vertically upwards and the air resistance has the same size during the journey. Which stage of the flight, upward or downward, takes the longer time?

Single Slit Diffraction

Light plays a key role in Physics. The study of the interaction between light and matter has caused many breakthroughs in in Physics. At a fundamental level, waves spread out after they pass through a gap. This process is called diffraction. When laser light passes through a narrow rectangular slit, and the pattern observed on a screen, most of the energy is found arriving on the screen in a bright area called the central maximum. This is flanked on each side by a dark area and then the pattern becomes brighter on each side at a location called the first order maximum. The first order maximum has an intensity of 4.7% that of the central maximum. Higher order maxima also form, the next two of which have intensity ratios of 1.7% and 0.83% respectively. Why doesn't all of the energy "land" in the central maximum? Why do we get higher order maxima?

HSC Physics Revision 2

Here are some more points to help students maximise their HSC Physics mark.

1. BCS Theory of Superconductivity This theory was developed by Bardeen, Cooper and Schrieffer in 1957. Below the critical temperature two electrons interact forming a bound pair known as a Cooper pair that does not collide with the atoms in the lattice and so there is no electrical resistance. As the first electron approaches the lattice the positive ions are distorted creating a concentration of positive charge that attracts the second electron. BCS theory states that the total momentum of a Cooper pair must be zero so the second electron must be moving in the opposite direction to the first. A common misconception is that the members of the pair must be close together. This is incorrect as at close distances the members of the pair would be repelled by their electrostatic repulsion. Cooper pairs do not lock together permanently. They change partners wihin the group of electrons and the net result is a movement through the material without collision. Any collision with the lattice or increase in temperature above the critical temperature destroys the superconducting state. Always draw a diagram showing the Cooper pair and the distorted lattice.
2. Quark Question What is the change in quark composition when a neutron decays?
3. Gravitational Potential Energy An apple is at rest on the surface of the Earth. What is the gravitational potential energy of the Earth-apple system? Is it zero?
4. Turning the Handle of a Generator When a the handle of a generator is turned with no resistor in the circuit it is "easy" to turn the handle. When a resitor is placed in the circuit it is "hard" to turn the handle. Why?
5. Proton and an Electron A proton and an electron are both released from rest in a uniform electric field. Compare the size of the momentum of each particle after each moves the same distance.
6. Fermi's Initial Experimental Observation of Nuclear Fission Do not confuse this dot point with Fermi's later work in 1942. Fermi was awarded the Nobel Prize in Physics for 1939 "for his demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reaction brought about by slow neutrons". Fermi did not discovery nuclear fission. He discovered the role of slow moving neutrons in triggering increased activity in uranium salts but did not connect this with the splitting of the uranium, which he later regretted.

Einstein and the Atomic Bomb

One of the greatest misconceptions in Science is that Einstein was the "father of the atomic bomb" or "the father of the nuclear age". As the NSW HSC Physics syllabus has the Option "Quanta to Quarks" where questions are asked on the Manhattan Project, students' answers sometimes wander into areas where Einstein is connected with the atomic bomb. What is this misconception based on? Einstein's famous equation 

E = mc²

is taught in Physics classes using problems involving nuclear power and so the mindset is formed that Einstein was the father of the nuclear age. This equation could equally be applied to calculate the loss in mass of a hot plate of food as it cools. Nowhere in Einstein's early papers is there mention of uranium or any technological application of it. Einstein stated that if a body gives off an energy L in he form of light its mass decreases by an amount L/c². This equation applies to any object and could be applied to wood or water. In 1907 Einstein stated his famous equation E=mc² giving the equivalence of mass and energy and it was not until 1932 that Cockcroft and Walton obtained the first experimental verification of it in a nuclear reaction. In 1939 Einstein feared that Germany may come into possession of nuclear weapons and at the instigation of many of the physicists who had left Europe he signed a letter to President Roosevelt warning of the possibility of a nuclear weapon and German interest in this area. Einstein's signing of this letter is sometimes interpreted as indicating that Einstein was involved in this research himself. Einstein had no connection with the atomic bomb project other than signing the letter. When it was later found that Germany did not make progress in nuclear research Einstein regretted signing the letter and stated "had I known that fear was not justified I would have not participated in opening this Pandora's box"

A Student's Guide to HSC Relativity 2

HSC Physics students have difficulty with length contraction problems in the Space section of the course. A graded set of tutorial questions is given below to assist students in length contraction problems. The questions are numerical, so that students can gain an understanding of these concepts by attaching numbers to them. 

Length. These questions compare the lengths Lo (the proper length or rest length of the object, being the length that is measured when the the observer is at rest relative to the object) and Lv (the length measured by an observer moving at a constant velocity v relative to the object).

L_v = L_o√(1 - v²/c²)

It is important to note that L is the length component in the direction of the relative velocity v. The relative motion causes a contraction in length Lv.

1. Fermilab's Tevatron particle accelerator is 6.3 km long. A proton moves at a constant speed of 2.3x10⁸ m/s through the accelerator. Determine the length of the accelerator in the reference frame of the moving proton.
2. An electron moving at 2.9x10⁸ m/s travels between two electrodes that are 6.5 cm apart in the reference frame of the laboratory. Find the distance between the electrodes in the reference frame of the moving electron.
3. Relative to an observer in the laboratory a metre stick has a velocity component of 0.97c parallel to its length. Determine the length of the stick according to the laboratory reference frame.
4. A straight rod lies along the x axis has a rest length of 35 cm. The rod moves along the x axis at a speed of 2.8x10⁸ m/s relative to the labortory reference frame. Determine the length of the rod in the laboratory frame of reference.
5. The length of a spaceship is contracted by 75% when measured from a reference frame. What is the speed of the spaceship relative to the reference frame?
6. Muons approach the Earth at a speed of 0.95c. The muon travels a distance of 200 km in its own reference frame. What distance does the muon travel in the refernce frame of the Earth?
7. A rocket X of rest length 80 m moves in a straight line at a speed of 0.92c relative to a rocket Y of rest length 60 m moving in the opposite direction. What are the length of Y according to X and the length of X according to Y?
8. A star is 6.5 light years from the Earth. A spaceship travels to this star at a constant speed of 0.93c. Determine the distance travelled by the spaceship according to its own reference frame.
9. A metre stick at rest in a reference frame S' makes an angle of 40° with the x' axis. A person in a different reference frame S determines this angle to be 60°. Determine the speed of S' relative to S.
10. A spaceship is moving away from the Earth at a speed of 0.87c. When the ship is at a distance of 3.6x10⁸ km from the Earth as measured in the Earth's reference frame a radio signal is sent to the spaceship from the Earth. Give the location of the spaceship in the Earth's reference frame when the signal is received.

Planck's Quanta

Physics students often ask how Max Planck's concept of energy quanta explains the shape of the blackbody radiation curve at all frequencies. How is it if we make the assumption E=hf we are able to avoid the prediction of classical wave theory that an infinite amount of energy is released at high frequencies (the ultraviolet catastrophe) ? 

What is Planck's quantum hypothesis?  

In 1901 Max Planck proposed that the vibrating atoms in the walls of hot objects can only have a discrete set of energy values. The energy of the oscillator is said to be quantised as it can only have certain quantities of energy given by the equation E=nhf where n=1,2,3.... The energy emitted by the atoms is in bundles of value hf where f is the frequency of oscillation and h is Planck's constant.

How does quantisation explain the shape of the blackbody radiation curve?

The first point to note is that not all of the atoms in the hot object are vibrating with the same energy. The number of atoms vibrating at energy E is proportional to e raised to the power of -E/kT  where k is Boltzmann's constant and T is the kelvin temperature. The exponential factor is a statistical factor that describes the spread of vibration energies throughout the object in much the same way as there is a range of heights of people in the population. The second point to note is that when the average vibration energy of an atom is calculated the energy quantisation rule (E = nhf) causes a geometric series to be formed that has a limiting sum and so the ultraviolet catastrophe is avoided. The result is that a very large number of atoms in the hot object vibrate at low frequencies, a large number of atoms vibrate at intermediate frequencies and a relatively small number vibrate at high frequencies. The energy-frequency graph therefore has a peak in the midrange (where most of the energy is released due to the large number of vibrating atoms in this range) and is small in height at the extremities (since a very large number of atoms vibrating at a low frequency gives a low energy output and a small number of atoms vibrating at a high frequency also gives a low energy output).

“He (Planck) believed that what the equations were telling him was that the packages of electromagnetic radiation could only be emitted or absorbed in amounts of hf, but that the radiation itself was a classical wave. A rough analogy might be with the relationship between a cash machine and money. The cash machine can only dispense money in multiples of $10-what you might call money elements.”

— Erwin Schrodinger and the Quantum Revolution by John Gribbin, Bantam Press 2012, page 74

Einstein's Photons

The photoelectric effect is a very frequent question in the HSC Physics examination in NSW. A certain amount of confusion has creeped into the examination responses of some students. Suppose that the frequency of the light being used to illuminate the metal is greater than the threshold frequency of this metal. Most students realise that electrons are released from the metal as the photon energy is greater than the work function of the metal. Now, let the frequency of the light be increased while keeping the intensity of the light constant. Does the number of electrons released per second increase, decrease or stay the same? Many students write that the number of electrons released per second stays constant as the intensity of the light does not change. But it does not. As we increase the frequency of the light (keeping its intensity constant) the number of electrons released per second decreases. Why??

Hannah's Sweets

Here is a probability problem with a difference. Normally, probability questions do not involve solving a quadratic equation to find the solution but this problem does. It comes from the Edexcel GCSE Mathematics A Paper 1 set in London in June 2015. This question was widely commented on when it appeared in the UK. With the trial examinations in most schools in NSW rapidly approaching it provides valuable practice for Extension 1 and 2 Mathematics students.

There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow.
Hannah takes at random a sweet from the bag. She eats the sweet.
Hannah then takes at random another sweet from the bag. She eats the sweet.
The probability that Hannah eats two orange sweets is 1/3.
What is the value of n?

10

From the Earth to the Moon

Jules Verne's classic science fiction story describes a cannon firing a projectile (with three people inside!) at the Moon. The story says that the projectile was fired so that it just reached the neutral zone (where the gravitational pull of the Moon on the projectile equals that of the Earth, called an unstable equilibrium point) when the Moon was at its closest point to the Earth in its orbit. The Moon's gravitational attraction then dominates and the projectile is captured by the Moon. If it is fired at a speed that just lets it pass through the neutral zone it reaches the Moon approximately 7 days after leaving the Earth. As the concepts in this problem are in the Space section of the NSW HSC Physics course, an interesting exercise is to apply the gravitational potential energy equation to this problem. Given the following data:

closest approach distance of the centre of the Moon to the centre of the Earth: 363,100 km
radius of the Earth: 6.38x10⁶m,
radius of the Moon: 1.74x10⁶m,
mass of the Earth:5.97x10²⁴kg,
mass of the Moon:7.36x10²²kg.

(i) What is the least speed at which the shell can be projected from the Earth if it is to reach the equilibrium point between the Earth and Moon? Neglect the orbital and rotational motions of each object.

(ii) If the projectile is slightly pushed from the equilibrium point towards the Moon, with what speed does it strike the Moon? Neglect the orbital and rotational motions of each object. (In Verne's story rockets were fired that put the shell in a lunar orbit)

(ii) If the orbital movement of the Moon is included, is the projection speed greater or less than in (i)?

greater, see K R Symon, "Mechanics", page 292

The Chudnovsky Brothers

It is often interesting in mathematics lessons to talk about curious characters who have done original things in mathematics. David and Gregory Chudnovsky are two brilliant brothers who will not be found in the index of most mathematics textbooks. In 1991 the brothers built a supercomputer in their apartment in Manhattan. They used mail order parts delivered in boxes, the building superintendent being unaware of what they were doing. They called their computer m-zero and claimed that it was just as powerful as a Cray supercomputer, the Cray costing $30 million and theirs $70,000. Why did they do this? Why fill their apartment with electrical leads and circuitry and raise its temperature to intolerable levels? To calculate pi. The brothers had a passion for mathematics and used their computer to calculate pi to two billion decimal places. Is it necessary to find this value to such high accuracy for everyday calculations? No. When we perform calculations our overall accuracy is determined by the least accurate number that we input. Most scientific constants are only known to at most 10 decimal places. The brothers were explorers determined to venture into new territory. It is only by pushing the limits of knowledge that new, and unexpected, discoveries, are made.

Physics in the Lab I

Doing experiments in a laboratory allows us to understand how the laws of physics work. We develop a knowledge of the quantity that we are measuring and how it depends on other factors. Experimental work involves making measurements and looking for patterns in the measurements. There are two concepts in experimental work that students have difficulty with. These are uncertainty (error) analysis and plotting data. As the NSW HSC syllabus no longer includes uncertainties in practical work many students arrive at first year university labs not fully prepared for the requirements of experimental work in both calculator skills and uncertainty analysis. The international examining boards, Cambridge International Examinations and the International Baccalaureate, still include questions with uncertainties in their examination papers. To help students learn uncertainty analysis a tutorial problem set is provided below.

1. A student drops a ball from the same height and measures the time of fall. Their measurements are 1.75s, 1.85s, 1.60s, 1.70 s and 1.71 s. Determine the average time of fall.
2. A student measures the dimensions of a desk top. The average value of the length was found to be 2.524 ± 0.004 m and the average value of the width was found to be 0.622 ± 0.004 m. Determine the perimeter and area of the desk top.
3. A student releases a ball from rest and measures the time it takes to fall to the ground. The average time was found to be 1.32 ± 0.08 s. Given that the height of release is 8.61 ± 0.05 m, determine the acceleration due to gravity.
4. A student measures the mass of a block as 117.56 ± 1.24 g. The volume of the block was measured as 22.67 ± 0.36 cm³. Determine the density of the block.
5. In a laboratory experiment a student measures the time of 10 oscillations of a simple pendulum of length 3.25 ± 0.03 m. Their time was 37.21 ± 0.86 s. Use this data to determine the acceleration due to gravity.
6. A dynamics trolley is moving along a smooth laboratory bench at a speed of 0.26 ± 0.03 m/s. The trolley accelerates at 0.84 ± 0.04 m/s² for 6.53 ± 0.08 s. Determine the distance travelled by the trolley.
7. Determine the volume of a right circular cylinder of radius 3.215 ± 0.025 m and height 7.512 ± 0.025 m.
8. Determine the volume of a sphere of radius 3.219 ± 0.038 m.
9. The density of plutonium is 19.8 ± 0.4 gcm^-3. Determine the radius in centimetres of a sphere of plutonium of mass 15.0 ± 0.5 kg.

10. When the radius r of the bob of a simple pendulum of length L is included in the calculation the period T of the small oscillations of the pendulum is given by the equation
    

    T = 2𝜋√(L/g + 2r²/(5gL))

    If L = 2.00 ± 0.02 m, g = 9.81 ± 0.03 ms^-2 and r = 10.0 ± 1.0 cm, determine the value of T.

11. When the angle of oscillation of a simple pendulum is not small, the approximate period of the oscillations is given by the equation
    

    T = 2𝜋√(L/g) (1 + 𝜽²/16)

    where L is the length of the string, g is the acceleration due to gravity and 𝜽 is the angle of release of the string from the vertical measured in radians. Taking L = 6.57 ± 0.05 m, g = 9.81 ± 0.04 and 𝜽 = 22 ± 3 ^∘, determine the value of T.

12. Determine the area of a triangle of sides 1.236 ± 0.015 m, 3.256 ± 0.023 m and 2.887 ± 0.023 m.

HSC Mathematics Reference Sheet

As term 2 starts in NSW it is worthwhile to take a breath and consider the impact of the new Reference Sheet that will be issued in HSC Advanced and Extension Mathematics examinations this year.  Will it help students in examination conditions or hinder them? This is a major change to mathematics education in NSW. Previously, students were not given formulae. Now formulae will be provided for topics in the Advanced and Extension 1 courses (formulae unique to Extension 2 topics are not provided). Many experienced teachers are advising their students to remember formulae (as in the past) and not fall into the trap of thinking that the sheet will cure poor mathematical understanding of concepts. Teachers have also commented that students are spending too much time flicking through pages looking for formulae when a simple application of important results, such as using the trig ratios of 30, 45 and 60 degrees, is required. As well, not all formulae are given, an example being the formula for the tan of the angle between two straight lines. It is important to reinforce that examination questions require the use of mathematical techniques and systematic working. Areas such as money payment problems in the Advanced course, which are usually poorly attempted, are not provided with formulae. Treat the Reference Sheet as an aid for the memory, not as a source of mathematical understanding. Once more, clearly show all working and be careful with basic algebra. This is the key to mathematical success.

Pi......From a Student's Perspective

Examiners in the HSC Advanced Mathematics (2 unit) course have commented that many students become confused when the constant 𝜋 is used in calculations. What is it about 𝜋 that causes problems? Students from an early age learn that 𝜋 is the ratio of the circumference of a circle to its diameter and so this reinforces the geometrical interpretation of 𝜋. However, 𝜋 can be involved in problems that have no connection with circles and this is when confusion in examination answers can arise. Some students convert an answer involving 𝜋 into revolutions or degrees just because of the  connection of 𝜋 to the geometry of a circle. Pi is a number, an irrational number, and should be treated like any other number in calculations. In the 2015 Mathematics examination question 15 c involved calculating the time when the volume of water in a pool first started to decrease given the rate of change of the volume. The answer was 2𝜋 seconds and many candidates lost marks by converting this into degrees.

“The fact that 𝜋, a purely geometric ratio, could be evolved out of so many arithmetic relationships- out of infinite series, with apparently little or no relation to geometry-was a never ending source of wonder and a never ending stimulus to mathematical activity”

— Mathematics and the Imagination by Edward Kasner and James Newman, G Bell and Sons 1951, page 78

Energy and Entropy

The word energy is widely used in everyday life. "I do not have the energy to exercise today" or "the world has an energy crisis" are familiar sayings. We use the word energy so much that it is taken for granted that we know what it means (?). The first law of thermodynamics tells us that energy can neither be created nor destroyed. When hot and cold water are mixed together the heat energy lost by the hot water equals the heat energy gained by the cold water. Another word in the vocabulary of physicists and engineers is entropy. Entropy is an indicator of the disorder of a system and the second law of thermodynamics states that natural processes tend to move toward a state of greater disorder (entropy) as time passes. The entropy of a system can never decrease. Students sometimes think that entropy is mysterious since it can be created, whereas energy cannot. When hot and cold water are mixed together the entropy of the hot water decreases and the entropy of the cold water increases by a greater amount. The total entropy of the mixture increases during the process. The mixing process created the entropy. The mixture does not naturally separate back into hot and cold components as this would involve a decrease in total entropy. Entropy is nature's arrow, always increasing, pointing in the direction in which systems evolve in time.

“If, when we observed a highly irreversible process (such as cooling coffee by placing an ice cube in it), we said, “That surely increases the entropy,” we would soon be as familiar with the word entropy as we are with the word energy.”

— Fundamentals of Classical Thermodynamics by Richard E Sonntag and Gordon J Van Wylen, 3rd edition page 232

What is Physics?

It is surprising that many students who are studying physics have never really thought of what physics actually is. A common answer given is that "physics is the study of the universe". On a smaller scale the common branches of physics are mechanics, thermodynamics, electricity and magnetism, relativity, quantum physics and nuclear physics. What do these areas have in common? Each of these areas studies something happening in nature, such as an apple falling to the ground, two magnets pushing on each other or the sun emitting light. To describe how nature behaves we use the language of nature, and this is mathematics. Combining all of these threads together we can define physics as the mathematical study of phenomena occurring in nature. The great physicist Paul Dirac commented on the association between mathematics and the natural world

“The mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by nature, but as time goes on it becomes increasingly evident that the rules that the mathematician finds interesting are the same as those which nature has chosen”

— Ian Stewart, Seekers after Truth and Beauty, Basic Books 2008, page 279

What is Mathematics?

Many students enjoy doing maths due to the satisfaction of agreeing with the answer when they turn to the back of the book. A sense of achievement comes from using thought processes to go through a series of steps to determine the correct result. Mathematics is a work-out for the mind just like a visit to a gymnasium is a physical work-out for the body. Harvard physicist Lisa Randall was drawn to mathematics "because it had definite answers....I liked the certainty of knowing which answers were right". Mathematics develops mental agility, logical thinking and systematic problem solving techniques. But what actually is mathematics? Algebra, calculus and geometry are branches of mathematics which operate using a common thread. Mathematics is the manipulation of patterns according to a defined set of rules or ideas. Three classic books that discuss the nature of mathematics  are What is Mathematics by Richard Courant and Herbert Robbins, Mathematics and the Imagination by Edward Kasner and John Newman and A Mathematician's Apology by G H Hardy. The Cambridge mathematician Hardy on page 84 gives the following insights into mathematics

“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”