Simple harmonic motion:graphical and analytical treatments

Characteristic features of simple harmonic motion

Exchange of potential and kinetic energy in oscillatory motion

Understanding and use of the following equations

Graphical representations linking displacement, velocity, acceleration, time and energy

Velocity as gradient of displacement/time graph

Simple pendulum

and mass-spring

as examples and use of the equations

Candidates should have experience of the use of datalogging
techniques in analysing mechanical and oscillatory systems

• Lots of natural systems exhibit SHM - bouncing, swinging things: atomic vibrations to bridges!
• you must always mention both characteristics of SHM:

1. Acceleration is proportional to the distance from a fixed point in its path
2. Acceleration is directed TOWARDS that point

• the top equation has acceleration (a) proportional to the negative value of the displacement (x) (i.e. proportional to the distance (magnitude of the displacement!) but opposite in direction - acceleration increases as it gets closer to the point we are measureing from. The constant of proportionality is linked to the frequency (f) of the oscillation.
• The middle equation relates the displacement (x) to the amplitude of the vibration (A) - you have got to be so careful with case of letter here!) as time progresses. This is a cosine relationship - at time zero x will be equal to the amplitude.
• The bottom equation gives the velocity at any displacement x. If it is at the extremities x=A therefore the velocity is zero. At the centre of the oscillation x=0 so v= 2pfA (maximum velocity) the +/- indicates the direction of travel.
• You should be able to sketch the graphs of displacement, velocity and acceleration against time for a system operating SHM. If you start at the point x at time t=0 then the displacement graph will be a sine curve (amplitude A), the velocity a cos curve (amplitude wA) and the acceleration a negative sine curve (amplitude w²A) - where w is 2pf
• The energy of the system is taken to be constant - made up of kinetic (max at x, where velocity is a max and zero at A because velocity at A is zero) plus potential which has the opposite characteristics of the kinetic.
• You should know how to skech and interpret displacement/time and velocity time graphs.
• The pendulum as an example of SHM - g is gravitational field strength and l is the length. The period will vary according to which planet you are on.
• The mass on a spring - m is the mass (kg) and k is the spring constant (stiffness of it - revise Hooke's Law!). This will not vary according to the planet it is on!
• A light gate can be used to detect the times that a swinging pendulum or bobbing mass on a spring cast a shadow over it. This is a more accurate and less tedious way of finding the period (and hence the frequeny) of an oscillation.

Free and forced vibration

Qualitative treatment of free and forced vibration

Resonance and the effects of damping

Examples of these effects from more than one branch of Physics e.g. production of sound in a pipe instrument or mechanical vibrations in a moving vehicle

Use examples eg Barton's Pendulums to explain free vibrations, forced vibrations and resonance.
State practical examples where resonance can have both positive and negative effects.
Explain how damping affects the energy in an oscillating system.

Natural Frequency
Draw graphs to show the effects of damping on an oscillating system.
See Physics Classroom - excellent explanations

Recall from GCSE the definitions of amplitude, frequency, wavelength, speed.
Explain the meaning of phase and ascertain the phase difference between 2 waves.
Calculate the path difference between two waves and relate this value to the wavelength in order to ascertain whether constructive or destructive interference has occurred.
Use the wave equation in calculations.

See here for 3-d interference

See here for Phasors

Longitudinal waves and transverse waves (examples including sound and electromagnetic waves)

Polarisation as evidence for the nature of transverse waves; applications, e.g. Polaroid sunglasses

State the difference between longitudinal and transverse waves giving examples of each.
Explain what polarised waves are and describe in detail an experiment to demonstrate the polarisation of light.
Practical: Polaroid pieces and light source

Try this link - can alter the phase and wavelength of waves and add them... also see longitudinal as well as transverse forms!

Superposition of waves

Stationary waves

The formation of stationary waves by two waves of the same frequency travelling in opposite directions (no mathematical treatment required)

Simple graphical representations of stationary waves, nodes and antinodes on strings and in pipes

Interference - see this interactive site

The concepts of path difference and coherence

Requirements of two source and single source double-slit systems for the production of fringes

The appearance of the interference fringes produced by a double slit system. This is a good interactive animation that illustrates diffraction through a single and double slit.

Diffraction: Appearance of the diffraction pattern from a single slit

The plane transmission diffraction grating at normal incidence

(Optical details of the spectrometer will not be required)

Derivation of d sin = n

Applications, e.g. to spectral analysis of light from stars

Capacitance Recall and use of

Do this section after electric fields
Carry out calculations using this equation.
Explain the effect of varying 2 of the variables on the 3rd.
Recognise the circuit symbols for polarised and non-polarised capacitors. Explain the practical differences between the two and suggest possible practical applications for each.
Practical: Show students electrolytic and non-electrolytic capacitors of widely varying capacitances (we have some very large caps. retrieved from an old amplifier!)

See animation on a variable capacitor's design.

See the animation on charging a capacitor.

Energy stored by capacitor - derivation and use of

and interpretation of area under a
graph of charge against p.d.

Graphical representation of charging and discharging of capacitors through resistors

time constant = RC

Calculation of time constants including their determination from
graphical data

Quantitative treatment of capacitor discharge

Candidates should have experience of the use of a voltage sensor and datalogger to plot discharge curve for a capacitor

Uniform motion in a circle where is angular speed

Centripetal force equation : recall and use of

Gravity, Newton's law, the gravitational constant G

Recall and use of

(Methods for measuring G are not included)

State and explain the significance of the terms in the equation including why the -ve sign is included.
Carry out calculations of gravitational force of attraction.
Compare the magnitudes of the fundamental forces.

Gravitational field strength g

Gravitational potential V

Graphical representations of variations of g and V with r

Motion of masses in gravitational fields

Circular motion of planets and satellites including geo-synchronous orbits

Compare and contrast radial and uniform fields in terms of gravitational field strength and gravitational potential NB this must include graphs

Coulomb's law, permittivity of free space e_o
Recall and use of

Using equation for gravitational force as a start 'derive' the expression for electrostatic force.
Explain meaning of permittivity, include absolute and relative values.
Explain action of dielectric in terms of capacitance (relate back to 13.2.1)

See Java Applet of field pattern

Electric potential V

Motion of charged particles in an electric field

Trajectory of particle beams

Similarities and differences between electric and gravitational fields

No quantitative comparisons required (....but they are good for your soul!)

Motion of charged particles in a magnetic field

F = BQv (field perpendicular to velocity)

Circular path of particles; application, e.g. charged particles in a
cyclotron

Magnetic flux density B, flux , flux linkage N

BA= (B normal to A)

Electromagnetic induction

Simple experimental phenomena, Faraday's and Lenz's laws

For a flux change at a uniform rate

Applications, e.g. p.d. between wing-tips of aircraft in flight

Recall Faraday's Law: that the magnitude of the induced EMF in a circuit is directlt proportional to the rate of change of flux linkage (or to the rate of cutting of magnetic flux).

Recall Lenz's Law: that the direction of the induced EMF is such that the current which it causes to flow opposes the change which is producing it.

Know that the induced EMF is therefore:

• directly proportional to the number of turns of wire
• proportional to the rate of cutting of flux (how fast the flux lines atr cut by the wire)
• produced in such a way as to make a current flow that has a magnetic field that opposes the field of the one causing the induction.

Mass and energy

Simple calculations on nuclear transformations; mass difference;

binding energy

Atomic mass unit, u

Conversion of units; 1u = 931.1 MeV

E = mc²

Appreciation that E = mc² applies to all energy changes

Graph of average binding energy per nucleon against nucleon number, A

Fission and fusion processes - see here for whole topic index

• know that the mass of nuclear particles when associated together in a nucleus (and therefore all matter!) is less than the sum of the mass of the component parts.
• the difference in mass between individual consitituents and the associated particles is called the 'mass difference'
• know that mass and energy are interchangable via the equation E = mc² (Einstein's equation).
• know that the conversion between mass (u) and energy (MeV) is possible via a shortcut in the databook that states the equivalence of mass and energy as:1u = 931.1 MeV
• Draw the graph of average binding energy per nucleon against nucleon number, A - including labelled axes with units and values on those axes!
• Recall that fission is the splitting into two of large nuclei and fission is the fusing (joining into one) of two smaller nuclei.
• Relate fission and fusion to the binding energy per nucleon graph to explain why the processes are energetically viable.

Induced fission Induced fission by thermal neutrons

Possibility of a chain reaction

Critical mass

Need for a moderator in thermal reactors

Control of the reaction rate

Factors influencing choice of material for moderator, control rods and coolant

Examples of materials

• a thermal neutron is a neutron that has energy in the infra red photon range.if U235 absorbs a thermal neutron (becomes U236) it is very unstable and will split into two (not usually equal) nuclei - a couple (on average 2 to 3) of free neutrons are also produced (these can go on to produce more fissions). The fragments are more stable (energetically viable reaction) and energy is released when this happens.The resulting nuclei are cassed frission fragments NOT daughter nuclei (that is the terminology in radioactivity!)The freed neutrons can go on to produce further fissions, but are usually to high energy to do this and need to be slowed down. This is done by a MODERATOR (moderates their speed!) such as graphite. It slows the neutron down by allowing multiple interactions with the carbon lattice without absorbtion of the neutron into the carbon atoms - graphite has a 'low cross section for neutrons'.A chain reaction is a reaction in which the instigator of the reaction is also produced as a product. It is therefore possible for the product of one reaction to go on to take the role of the reactant in a future reaction.Each fission produces neutrons that could go on to produce further fissions so the more atoms you have (greater mass of sample) the more likely that the reaction will continue in a chain reaction. But those neutrons are produced isotropically (equally in all directions) - the production direction is random, so an atom on the surface could well shoot off a neutron out ofthe Uranium mass and no fissions would then occur from them. The bigger the surface area of the Uranium sample the more likely that neutrons will be sent out and not be able to make more fissions. As mass increases so does volume and surface area of the sample. A very small mass will have a larger surface area relative to its size than a bigger one so a chain reaction is less likely (greater proportion of its atoms will be on the surface). There is therefore a minimum mass that allows a chain reaction to occur. This is called the critical mass - it has a critical mass/surface area ratio below which a chain reaction is not viable.As 2/3 neutrons are produced each time a nucleus of Uranium splits the energy produced by reaction would escalate by a factor of about 2/3 at each stage. This would be uncontrolled acceleration of the reaction and be very dangerous (bomb).Control rods of cadmium or boron can be inserted into the reaction vessel to maintain the energy production at the required level. These have a 'high cross section for neutrons' - they absorb the neutrons, taking them out of the reaction preventing further fissions occuriing. The deeper the rods are inserted into the vessel the faster the rate at which energy is being produced will be diminished (more surface area of absorber - more absorbtion)Moderator materials are chosen for low cross section for neutrons - don't absorb neutrons - interact to take kinetic energy from them instead.Control rods are made of materials that absorb neutrons effectively - have a high cross section for neutrons.
• Coolant (eg. water or CO₂) needs to have a high specific heat capacity so that a large amount of heat energy can be absorbed without it getting too hot

Safety aspects:

Fuel used, shielding, emergency shutdown

Production, handling and disposal of active wastes

Uranium is an alpha (and gamma!) emitter - dust is very dangerous - highly localised ionisation in the body from alpha results in a high risk of cancer and mutaion in reproduction.

All forms of radiation are produced by the cocktail of fission fragments so shielding must be thick concrete (lead oo expensive!)

Control rods contain enough absorbing material to completely shutdown the reactor - absorb so many neutrons that the chain reaction is stopped.

Fission fragments are radioactive isotopes of many elements - variety of half lives and type of radiation produced.

Neutron absorbtion by an atom creates a radioactive isotope of that element - which then decays into something else! So the reactor itself and instruments used within the reaction vessel become highly radioactive.

You should know about the three categories of waste (low, intermediate and high) due to their half lives (how long they will be dangerous for) and their activity, how they are dealt with (stored, monitored).

Also revise safety when dealing with radioactive materials for handling of active waste.

Artificial transmutation

Production of man-made nuclides and examples of their practical applications, e.g. in medical diagnosis

Neutron absorbtion by an atom creates a radioactive isotope of that element - which then decays into something else, so we can put copper into a reactor - bombard it with neutrons and make radioactive versions of it (other isotopes). When the radioactive copper atoms decay they change into another element - we have done it (not natural - artificial), transmutation (changing into something else!

All of the positron emitters have been produced in this way (decay from artificially created isotopes)... isotopes that we would not find naturally on our Earth have daughters that become positrom emitters.

You can also bombard atoms with alpha particles, protons and deuterium (Hydrogen2) to create artificial nuclei.

Nuclear medicine uses radioactive atoms as tracers (with gamma camera). The tracer must be made of atoms that the body uses in a particular biological way. 'Labelling' molecules with radioactive atoms enables functions to be monitored. These often do not occur in nature or are rare. Artificial transmutation is an excellent way of obtaining them.