Velocity-Time Graph

Question 16 in the 2016 HSC Mathematics examination involved calculating the distance travelled by a particle given its velocity as a function of time. The velocity during the first second of the motion was negative. This trapped many students. To find the total distance travelled we must add up the absolute values of all of the areas under the v-t graph. Be careful! If you have done it correctly the answer is 10-4ln2. If you obtain 14-12ln2 this is the change in displacement or the final position of the particle measured from the starting point; this is the length of the arrow from the starting point to the finishing point and is not the distance travelled which is the total amount of ground covered.

Relativity

Question 5 in the 2003 HSC Physics examination was a multiple choice question on time dilation and length contraction. Unfortunately, no correct answer was given as an alternative. Students doing revision often ask about this question. As the mark for the question was apparently not included in the final mark it could probably be ignored. However, an answer involves taking into account the travel time of the light coming from the Earth as the question says "when seen from the astronaut's spaceship"...seen implying making an observation using light .  From the point of view of the astronaut in the spaceship the Earth is moving away at 0.8c. The time for the journey in the reference frame of the spaceship is 10 years. The distance of the journey in the reference frame of the spaceship is 8 light years and so a ray of light would take a time interval of 2 years in the spaceships reference frame to reach the spaceship. To determine the corresponding time interval shown by the clock on the Earth we solve the time dilation equation for t0 putting tv as 2 and v as 0.8c. This is because the astronaut considers the Earth to be moving away carrying its clock with it. This works out to be 1.2 years.

Simple Pendulum

The simple pendulum is one of the oldest Physics demonstrations and examination questions. A simple pendulum consists of a mass tied to one end of a string, the other end of which is fixed, and the mass is allowed to swing freely in a vertical plane. The important physical concept involved is energy. At any point of its motion the energy (meaning the "total" energy) of the pendulum is constant, provided frictional forces are negligible. Energy is said to be a constant of the motion. In Physics problems we always look for constants. Constants allow us to determine many properties of the motion of a system. Here is a list of some pendulum problems that students usually find difficult.

1. Determine the magnitude of the acceleration of the mass when it is at the lowest point of its swing. Is it zero? Is it g?
2. What is the direction of the acceleration vector of the mass at the lowest point of its swing?
3. Determine the magnitude of the acceleration of the mass at the highest point of its swing. Is it zero?
4. Imagine that a simple pendulum of mass m and length L is set moving so that it just reaches the vertical position over the point of support. Determine the energy of the pendulum in terms of g, L and m. Neglect frictional forces.[2.5mgL]
5. Imagine that the mass is set moving and the string becomes slack before it reaches the vertical position. The mass then falls on a path that passes through the point of support. Determine the energy of the mass in this situation in terms of g, L and m. Neglect friction forces. [1.86603mgL]
6. As in question 5 but now the path of the falling mass passes through the lowest point of the swing of the pendulum.[1.75mgL]
7. As in question 5 but now the path of the falling mass passes through the horizontal through the point of support at a distance L from the point of support. [2.29904mgL]

Motors and Generators

The Motors and Generators in the NSW government syllabus is usually answered poorly in Year 12 examinations. What are the reasons for this? Firstly, magnetic fields are abstract things....we cannot see them but we can measure their effects when we do experiments. Secondly, the direction of the magnetic force vector acting on a current carrying conductor is perpendicular to both the magnetic field vector and the current vector and this presents challenges in thinking. Finally, students' exam responses sometimes become confused due to a lack of understanding of the basic terms used in this topic. Here is a list of some of the basic facts in this topic.

1. Magnetic field and magnetic force are not the same thing.
2. Current is the rate of flow of charge through a conductor.
3. Current is measured in ampere (A). Charge is measured in coulomb (C)
4. The potential difference between two points is the work done in moving a +1 C charge between the points.
5. Potential difference is measured in volts (V).
6. Power is the rate at which work is done.
7. Work is measured in joules (J). Power is measured in watts (W).
8. Current direction is the opposite to the dirction of electron flow.

9. A current carrying conductor experiences a magnetic force when it is placed in a magnetic field.

10. When a conductor moves through a magnetic field a potential difference is induced across the conductor.

11. When a current carrying conductor is placed in an external magnetic field the interaction of the external magnetic field and the magnetic field of the conductor does not exert a force on the wire.

The Value of Physics

For a moment, let us put to one side the value of Physics as a science in its own right that describes the natural world. By studying Physics we pick up skills that flow over into other subjects. A course in Physics teaches us problem solving techniques and ways of thinking that can be applied in other areas. Physics teaches us how to collect and analyse data and then how to explain our measurements in terms of fundamental laws. Physics is highly rated as a "facilitating" subject by the major universities. Many university courses require particular subjects to be studied in the final years of secondary schooling. These are the facilitating subjects. The top 24 universities in the UK, known as the Russell group, publish a document called Informed Choices, giving information for UK university entry. This document states that the top three facilitating subjects in preparation for university study are Mathematics/Further Mathematics, English Literature and Physics. 

Lenz's Law

A very important law that connects electricity and magnetism is Lenz's law. When written as an equation, this law involves two mathematical concepts, a negative sign and a first derivative. Students can perform mathematical operations involving both of these concepts. The difficulty arises, and this happens in many equations, when we try to put a physical interpretation on what the negative sign and the derivative are trying to "tell us". In Physics we need insight into what an equation is saying. To a physicist an equation is a connection between quantities that have physical meaning; an equation describes how nature behaves. As Lenz's law is a common topic in HSC, IB, Cambridge International and first year university examination papers it is worthwhile to compile a list of the statement of Lenz's law as given by the current popular Physics textbooks to see how experts interpret it.

1. Halliday, Resnick and Walker Fundamentals of Physics (14th edition) An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current

2. Serway and Jewett Physics for Scientists and Engineers (8th edition) The induced current is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop.

3. Sears, Zemansky, Young and Freedman University Physics (14th edition) The direction of any magnetic induction effect is such as to oppose the cause of the effect

4. Knight Physics for Scientists and Engineers (4th edition) There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in flux.

5. Cutnell and Johnson Physics (9th edition) The induced emf resulting from a changing magnetic flux has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes the original flux change.

6. Tipler and Mosca Physics for Scientists and Engineers (7th edition) The induced emf is in such a direction as to oppose, or tend to oppose, the change that produces it.

Which do you think is clearest?

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.