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Advancing Physics A2
Chapter 10 Creating models
John Mascall
Student Notes
The King’s School, Ely
August 2009
Assessable learning outcomes for Chapter 10
Candidates should demonstrate evidence of:
1. knowledge and understanding of phenomena, concepts and relationships by describing and
explaining cases involving:
(i) capacitance as the ratio C =Q/V;
(ii) the energy on a capacitor E = ½QV;
(iii) the exponential form of the decay of charge on a capacitor as due to the rate of removal of
charge being proportional to the charge remaining;
dQ
Q

dt
RC
(iv) the exponential form of radioactive decay as a random process with a fixed probability, the
number of nuclei decaying being proportional to the number remaining;
dN
 N
dt
(v) simple harmonic motion of a mass with a restoring force proportional to displacement such
that
d2 x
k


x
m
dt 2
(vi) kinetic and potential energy changes in simple harmonic motion;
(vii) free and forced vibrations, damping and resonance
(qualitative treatment only);
2. scientific communication and comprehension of the language and representations of physics
by making appropriate use of the terms:
(i) for a capacitor: time constant τ;
(ii) for radioactive decay: activity, decay constant λ, half-life T½, probability, randomness;
(iii) for oscillating systems: simple harmonic motion, period, frequency, free and forced
oscillations, resonance;
by expressing in words and vice-versa:
(iv) relationships of the form
dx
  kx , where rate of change is proportional to amount
dt
present;
by sketching, plotting from data and interpreting:
(v) decay curves, plotted directly or logarithmically;
(vi) energy of capacitor as area below a Q–V graph;
(vii) energy of stretched spring as area below a force–extension graph;
(viii) v–t and a–t graphs of simple harmonic motion including their relative phases;
(ix) amplitude of a resonator against driving frequency;
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3. quantitative and mathematical skills, knowledge and understanding by making calculations
and estimates involving:
(i) calculating activity and half life of a radioactive source from data, T1/2; =ln 2/λ
(ii) solving equations of the form
dN
 N by iterative numerical or graphical methods
dt
(N = N0 exp(–λt) as the analytic solution);
(iii) calculating time constant τ of a capacitor circuit from data; τ = RC; Q = Q0 exp(–t/RC);
(iv) solving equations of the form
dQ
Q
by iterative numerical or graphical methods;

dt
RC
(v) C = Q/V, I = ΔQ/Δt, E = ½QV , E = ½ CV2;
(vi) T = 2 π √(m/k) , with f = 1/T ; and analogous equations such as that for the simple
pendulum;
(vii) F = kx ; E = ½kx2;
d2 x
k
(viii)solving equations of the form
 x
2
m
dt
by iterative numerical or graphical methods;
(ix) x = A sin 2πft or x = A cos 2πft;
(x) Etotal = ½ mv2 + ½ kx2
A Revision Checklist for Chapter 10 can be found on the Advancing Physics
CD-ROM.
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Ch 10.1 What if…?
Learning outcomes

Models are simplified descriptions of reality, obeying definite rules.

Exponential changes are ones in which the rate of change of a quantity is
proportional to that quantity.

Radioactive decay is exponential, if the random nature of the decay is smoothed
out, such that
dN
 N
dt

The half-life of the radioactive decay is the (constant) time taken on average for the
number of decaying nuclei to be halved.
We start with some modelling using WorldMaker on the Advancing Physics CD-ROM.
WorldMaker enables you to make decisions about how to simplify real situations, and
come to appreciate the flexibility, which enables models to imitate such a wide range of
physical phenomena.
You may wish to look at ‘How to… Use this CD-ROM…WorldMaker…Getting started with
WorldMaker’ on the CD-ROM before carrying out the tasks below.
All students should work through:
Activity 10S Software Based 'Models of forest fires and percolation'
using File 10L Launchable File 'Forest fire models' and File 20L Launchable File
'Percolation models'
The two models only differ in what the icon objects look like, and what they are called.
Neither affects the way the model works, which is exactly the same in both.
If there is time you can work through:
Activity 20S Software Based 'Models of rabbit populations'
using File 30L Launchable File 'Rabbit numbers'
The model is used to introduce the idea of exponential change. It shows that rabbit
populations can grow exponentially if birth rate exceeds death rate, until space runs out
and that rabbit populations can fall exponentially if death rate exceeds birth rate, until there
are none left. In both case, exponential change occurs when the rate of change in
numbers is proportional to the number present.
To get started you should launch WorldMaker from the programme menu. The model files
needed are fires.mb, percol.mb and rabbits.mb which can be found in the ‘worlds’ folder of
WorldMaker.
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Exponential change and radioactive decay
You must understand what is meant by ‘half-life’ from work covered at GCSE. It is
important to appreciate that radioactive substances do not last forever and that some
decay very quickly whereas others seem to have an infinite life.
Notes:
At this stage you will see a demonstration of a radioactive nuclide (protactinium) decaying
over time.
Activity 30P Presentation 'Observing exponential decay: Radioactivity'
If a data logger is not used, the data can be analysed using Excel having previously
subtracted the background count. By using Excel to fit a smooth (exponential) trendline,
the contrast with the experimental values becomes apparent; the effects of randomness
should be clear with the experimental data. Measure the half-life from the trendline.
Notes:
Question
If the fraction of isotope remaining after three half-lives is (1/2)3 = 1/8 = 0.125, what
fraction will remain after 1.5 half-lives?
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The part played by probability can be illustrated by an analogue class experiment using
dice.
Activity 50E Experiment 'A model of radioactive decay using dice'
It is useful to use the graphs to compare the experimental values with the predicted ones.
Again, the experimental graph shows the effects of randomness.
We can predict the outcome of each throw by using the equation ∆N = - (1/6) N.
N represents the number of dice showing a ‘6’ before the throw, and ∆N represents the
change in that number as a result of the throw. ∆N is negative if we are dealing with a
decaying population. The 1/6 represents the probability that a die will fall with the number
‘6’ facing upwards.
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If we start with a population of 120 dice, ∆N after the first throw will be - (1/6) × 120 = - 20.
If all these dice are removed, the number remaining at the start of the next throw should be
N + ∆N = 120 - 20 = 100. However, because the process is random, the actual number of
dice removed on each throw may not be exactly as predicted. As with real radioactive
decay, the effect of randomness becomes apparent if the numbers used are relatively
small.
Use the theoretical model given above to complete the table below for the dice
experiment. The table has been started for you.
Throw number
Number of dice at
start of throw.
0
100
1
80
Change in number of Number of dice
dice each throw
remaining after each
throw
- 20
80
2
3
4
5
It is useful to see a short demonstration of decay using WorldMaker model.
Activity 60S Software Based 'Models of radioactive decay'
Using File 50L Launchable File 'Single decays in WorldMaker'
Radioactive decay can be modelled by creating an object (‘the unstable nucleus’) which
has, at every moment the computer model is running, a fixed probability of changing into
another object (‘the stable nucleus’).
Developing a theoretical model of radioactive decay
The number of nuclei decaying in a time t = pN where p is the probability of decay and N is
the number of undecayed nuclei at that time.
p =  t where  is the decay constant (probability of decay in a fixed time interval).
[Note that p depends on the decay constant and the time interval as you might expect.]
Activity a = number decaying per second
= pN = ( t)N = N
t
t
The activity is the positive value of dN/dt where dN/dt = -  N.
Note that this is a differential equation in which the rate of change of N is directly
proportional to N.
Notes:
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Key points:
a = N
dN = -  N
dt
 (decay constant) is the probability of decay per unit time. If the unit of time is the
second,  will have units of s-1.
The unit of activity is the becquerel (Bq)
Exercise
(a) Using one set of axes, sketch idealised graphs of N against t for two isotopes starting
with identical numbers of undecayed nuclei but with very different half lives. Annotate your
graphs to indicate the difference in the decay constant in each case.
(b) 1 g of radium-226 has an activity of 3.7  1010 Bq. [This activity was formally known as
a curie.] Calculate the decay constant for radium. The Avogadro constant is 6.02  1023
mol-1 so 226 g of radium-226 will contain 6.02  1023 undecayed nuclei.
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(c) A radioactive source used to generate electrical power has an activity of 5  1010 Bq
and emits particles of energy 5 MeV. Assuming that all of the energy released from the
source is used to generate the heat used to produce the electricity, calculate the maximum
theoretical electrical power output of the source. You might like to think why this power
output cannot be achieved in practice.
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Display Material 10O
OHT 'Smoothed out radioactive decay'
Smoothed-out radioactive decay
Actual, random decay
N
N
t
t
time t
probability p of decay in short time t is proportional to t:
p =  t
average number of decays in time t is pN
t short so that N much less than N
change in N = N = –number of decays
N = –N
t
 N = –pN
 N = –N t
Sim plified, smooth decay
rate of c hange
= slope
= dN
dt
time t
Consider only the smooth form of the average behaviour.
In an interval dt as small as you please:
probability of decay p =  dt
number of decays in time dt is pN
change in N = dN = –number of decays
dN = –pN
dN = – N dt
dN = –N
dt
Actual, random decay fluctuates. The sim plified model sm ooths
out the fluctuations
Compare actual random decay with the simplified smooth decay.
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Exercise
(a) Sketch an idealised graph of number of
nuclei remaining against time t for an
isotope that starts with 10 000 undecayed
nuclei and has a decay constant of 0.1s-1.
You will see later that the half-life will be
approximately 7 s.
(b) Sketch an idealised graph of activity a
against time t for the same isotope as
above. Use the same scale as above
for the time axis.
Radioactive clocks.
The text discusses use of radioactive decay (radiocarbon dating) to track the ancient
origins of farming (p 7).
There is an important link between half-life and decay constant:
t1/2 = (ln 2) = 0.693


[You will see in Section 10.2 that the differential equation dN/dt = -  N has a solution
N = N0 e-t. When N = N0/2, N0/2 = N0 e-t 1/2 and so e-t 1/2 = ½. By taking logs of both sides
we have  t 1/2 = ln 2 where t 1/2 represents the half-life.]
Notes:
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Exercise
(a) In an exercise above you should have found that the decay constant of radium-226 is
approximately 1.4  10-11 s-1. Calculate the half-life of radium-226.
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(b) Why would a direct measurement of the half-life of radium-226 be rather tricky?
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Try this
The activity of a radioactive isotope falls to 1/8th of its initial value in 24 days.
Without using any special formulae, find the number of half-lives that have elapsed and
hence calculate the half-life of the isotope.
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Did you get 8 days for the half life? A more general set of formula is given below that will
enable you to handle questions where the time elapsed is not a whole number of half-lives.
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Display Material 20O
OHT 'Radioactive decay used as a clock'
Clocking radioactive decay
Activity
Half-life
N0
number N of
nuclei halves
every time t
increases by
half-life t 1/2
N0 /2
slope = activity =
dN
dt
halves every half-life
N0 /4
N0 /8
t 1/2
t 1/2
t 1/2
t1/2
t 1/2
time t
time t
Radioactive clock
In any time t the number N is reduced by a constant factor
Measure activity. Activity proportional to number N left
In one half-life t 1/2 the number N is reduced by a factor 2
Find factor F by which activity has been reduced
In L half-lives the number N is reduced by a factor 2 L
Calculate L so that 2 L = F
(e.g. in 3 half-lives N is reduced by the factor 23 = 8)
L = log2F
age = t 1/2 L
◦ In one half-life, N falls by a factor of 2.
◦ The number of nuclei is reduced by a factor F which equals 2L in L half-lives.
◦ Since F = 2L we can use L = log2F to find L.
◦ The time elapsed would then be t1/2 L.
◦ If time elapsed and L are both known, t1/2 can be calculated. Note that L does NOT have
to be a whole number of half-lives.
F = 2L
L = log2F
age = t1/2 L
In the example above, F = 8 so L = log2 8 = 3.
The time elapsed was t1/2 L = t1/2  3 = 24 days.
This gives the same value of 8 days for the half-life.
Question
A sample of iodine-131, with a half-life of 8.04 days, has an activity of 7.4 × 107 Bq.
Calculate the activity of the sample after 6 weeks.
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Ch 10.2 Stocks and flows
Learning outcomes

The charge on a capacitor of capacitance C at potential difference V is Q = CV so
that C = Q / V and V = Q / C. The unit of capacitance is the farad.

The differential equation for exponential change of a quantity Q is
dQ
 kQ
dt
with exponential growth if k is positive and exponential decay if k is negative.

The differential equation for discharge of charge Q on a capacitance C through
resistance R is
dQ
Q

dt
RC

The solution of the differential equation for discharge of a capacitor is
Q
 e -t/RC
Q0
N
 e -t
The corresponding solution for radioactive decay is N 0



The time constant RC is the time for the charge to reduce to
1 / e ≈ 0.37 of its former value.
The half-life of radioactive decay t1/2 is equal to loge 2 ≈ 0.693 of the
time constant 1 / λ.
The energy stored on a capacitor = ½ QV = ½ CV2 = ½ Q2 / C.
Capacitors and exponential change
We introduce charging and discharging of capacitors using electron flow ideas. The
analogy with water flows is useful.
A short introductory demonstration shows the structure of a capacitor (two metal sheets
separated by an insulating material) and helps to establish ideas about charges
+Q and –Q on the plates. More p.d. gives more charge and the capacitor stores more
energy.
Activity 90D Demonstration 'Super-capacitor'
Notes:
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The very first part of the following class experiment explores charging and discharging of a
capacitor.
Activity 100E Experiment 'Charging and discharging capacitors'
Summary of charging and discharging :
You can now use a coulombmeter to make direct measurements of charge Q, and the way
it varies with applied voltage V. This also establishes the idea of capacitance.
Activity 110E Experiment 'Measuring the charge on a capacitor'
Summary:
A capacitor acts as a store of opposite charges that are kept separated.
The net charge stored is (+Q) + (-Q) = 0.
Because these charges have been pulled apart, the capacitor stores energy.
QV
Q = CV so capacitance C = Q / V
The unit of capacitance is the farad.
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Display Material 30O
OHT 'Analogies between charge and water'
Stores of water and electric cha rge
dam filled
with water
pressure difference
increases as amount
of water in dam
increases
Electric charge
conducting plates with
opposite charges
concentrated on them
define capacitance:
C=
Q
V
charge stored per volt
–Q +Q
potential difference V
increases as amount of
charge stored increases
to calculate Q or V:
Q = CV
V=
Q
C
units:
charge Q
potential difference V
capacitance C
coulomb C
volt V
farad F = C V –1
Capacitors store electric charge. The larger the capacitance the larger
the charge stored at a given potential difference
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Storing charge / storing water
Potential difference depends on the quantity stored in both cases.
We can get a feel for the exponential decay of charge on a capacitor from the water
analogy. The exponential nature of the decay can be confirmed by a brief demonstration of
Activity 130E Experiment 'Analysing the discharge of a capacitor'
Exponential water clock
what if....
volume of water per second flowing
through outlet tube is proportional
to pressure difference across tube,
and the tank has uniform cross
section?
height h
volume of water V
pressure difference
p across tube
flow rate f =
dV
dt
fine tube to restrict flow
Pressure difference
proportional to height h.
Constant cross section so
height h proportional to volume
of water V:
pV
Rate of flow of water
proportional to pressure
difference:
f=
dV
p
dt
F low of water decreases water volume
rate of change of water volume proportional to water volume:
dV
 –V
dt
Time to half empty is
large if tube res ists
flow and tank has
large cross sec tion
t
Water level decays exponentially if rate of flow proportional to pressure
difference and cross section of tank is constant
Water running out
An exponential change arises because the rate of loss of water is proportional to the
amount of water left.
Page | 15
Exponential decay of charge
Wh at if....
current flowing through resistance is proportional to potential
difference and potential difference is proportional to charge
on capacitor?
capacitance C
charge Q
current I
resistance R
potential difference, V
current I = dQ/dt
Potential difference V
proportional to charge Q
Rate of flow of charge
proportional to potential
difference
V = Q/C
I = dQ /dt = V/R
flow of charge decreases charge
rate of change of charge proportional to charge
dQ /d t = –Q /RC
tim e for half charge
to decay is large if
resistanc e is large
and capacitance is
large
Q
t
Ch arge d ecays ex ponen tially if c urrent is propo rtional to po tential
differen ce, and capacitance C is co nstant
Charge running out
An exponential change arises because the rate of loss of charge is proportional to the
amount of charge left.
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This leads to the differential equation:
dQ = -Q
dt
RC
which is similar to the differential equation established in Section 10.1 for
radioactive decay.
We can use the equation dQ/dt = kQ as the general form of the differential equation for
exponential change, where Q can represent any quantity. If k is positive we have
exponential growth and if k is negative we have exponential decay.
For charge in a RC circuit, you should appreciate that the current in a circuit containing a
capacitor is quite different from the steady flow of charge through a resistor in a direct
current circuit.
Charging: As charge accumulates on the surfaces of a capacitor being charged, a
potential difference is developed so as to oppose the flow of charge. The result is not a
constant flow but one which gradually decreases as charge accumulates.
Discharging: When a capacitor discharges and charge flows in the opposite direction, and
charge falls as the p.d. across the capacitor falls. Charge, current and p.d. all follow this
exponential pattern.
The following equations describe the discharge of a capacitor:
Charge Q = Q0 e-t/RC
Voltage V = Q/C and C is constant, so V = V0 e-t/RC
Also the current I through resistor R is I = V/R which gives I = I0 e-t/RC
Exercise
You will soon see a repeat of the demonstration of Activity 130E to test the equation
V = V0 e-t/RC where R = 100 kΩ, C = 500 μF and the initial p.d. V0 is 6.0 V. It is worth
calculating the values you expect to obtain so that you gain some practice in using the
exponential equation.
(a) Show that RC = 50 s.
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(b) Use your calculator to complete the table below.
t/s
0
10
t/RC
0
0.2
e-t/RC
1
0.82
6.0
4.9
V0 e-t/RC
20
30
40
50
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The time constant RC (measured in
seconds) is the time for the voltage,
charge or current to fall to 1/e or 0.37 of
its former value.
(c) Use the value of V at 50 s from the table above to show that the time constant is the
time for the voltage to fall to 0.37 of its original value.
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You will now see a demonstration of capacitor discharge using the values used in the
calculation. Note how close the experimental values are to those obtained by calculation.
Question
Why are the experimental and calculated values not exactly the same?
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You may recall from work in Chapter 1 that:
1000 = 103 so log101000 = 3.
Likewise, if y = ex then loge y = x. The exponential function is the inverse function to the
natural logarithm.
If x increases linearly then y will increase exponentially but loge y will increase linearly.
Plotting the log of an exponential
function will produce a straight
line graph.
In our case, V = V0 e-t/RC so lnV = ln V0 –t/RC.
This means that plotting lnV against t will give a straight line graph, providing a useful
method for finding the time constant RC of a circuit. The gradient of the lnV against t graph
is –1/RC.
The data generated from the demonstration above can be used to check this. This is best
done using Excel.
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Notes:
You should now explore the discharge of a capacitor yourself.
Activity 130E Experiment 'Analysing the discharge of a capacitor'
The exponential nature of the discharge is explored using a datalogger and Insight
software. Note that the experiment involves a measurement of decay constant which is
1/RC.
Modelling capacitor discharge will be explored using either:
Activity 140S Software Based 'Modelling the Euler algorithm graphically'
using File 90L Launchable File 'The step by step integration of dN/dt = –kN'
Only an approximate solution to the differential equation is obtained.
or
Activity 150S Software Based 'Stepwise through decay'.
Here you will model decay using Modellus.
If time we will support the work of this section with
Activity 180S Software Based 'Capacitor discharge'
which uses File 120L Launchable File 'Capacitor discharge'
In this model we can see the effect on the discharge of changing RC.
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Display Material 40O
OHT 'Half-life and time constant'
Radioactive dec ay times
N/N 0 = e–t
dN/dt = – N
N0
N 0 /2
N 0 /e
0
t=0
t = t1/2
t = time constant 1/
Tim e con stant 1/
at tim e t = 1/
N/N 0 = 1/e = 0.37 approx.
t = 1/is the time constant of the decay
Half-life t1/2
at time t1/2 n umber N becomes N 0/2
N/N 0 =
In
1
2
1
2
= – exp(– t1/2)
= –t 1/2
t 1/2 =
ln 2 0.693
=


In 2 = log e 2
The half-life t 1/2 is related to the decay constant 
Throughout this section we have treated charge as a continuous variable from which
random fluctuations are absent. Although the motion of electrons in the circuit will lead to
random fluctuations in current, the number of charge carriers is so large that the variation
appears continuous and smooth. The flow of charge is in fact a statistical average, subject
to random variation, but these effects are too small to be noticed.
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Energy stored by capacitors
Capacitors can be used as reservoirs of energy to be released on demand, as in a camera
flash gun, or for use as memory backup stores of energy if a battery or the mains should
fail.
You may see a short demonstration of discharge of various capacitors involving an
obvious release of energy (e.g. to produce a spark; to light a mains lamp). CARE IS
NEEDED whilst charging capacitors to high voltages. It is important to make sure that
capacitors are fully discharged after use.
Energy stored in a capacitor can be found from the area under a Q – V graph:
Display Material 50O
OHT 'Energy stored on a capacitor'
Energy stored on capa citor = 1 QV
2
add up strips to get triangle
capacitor discharges
V0
V1
V2
energy
E
delivered
= V 1 Q
energy
E
delivered
= V2 Q
Q
V0 /2
1
energy = area = 2 Q 0 V 0
Q
Q0
charge Q
charge Q
Energy delivered at p.d. V when a small charge Q flows E = V Q
Energy delivered = charge  average p.d.
Energy E delivered by same charge Q falls as V falls
Energy delivered =
Capacitance, charge and p.d.
C = Q/V
1
2
Q 0 V0
Equations for energy stored
E=
1
2
Q 0 V0
Q 0 = CV0
E=
1
2
CV 0 2
V0 = Q 0 /C
E=
1
2
Q 0 2 /C
Question
An electronic flash gun has to provide a power of 20 kW for about 2 ms. This energy is
obtained by discharging a 50 μF capacitor through a circuit with a time constant of 2 ms.
(a) Calculate the energy stored in the capacitor.
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(b) Calculate the voltage required to charge the capacitor.
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(c) Calculate the resistance of the discharge circuit.
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(d) Explain why the flash gun takes much longer than 2 ms to charge.
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Please be aware that the discharge is exponential and so these calculations are very
approximate.
The next experiment is likely to be set up as a demonstration but you will be invited to take
readings yourself.
Activity 190E Experiment 'Energy stored in a capacitor and the potential difference across
its plates'
This experiment relies on the heating effect of the current which flows when the capacitor
is discharged resulting in a measurable rise in temperature. The rise in temperature is
assumed to be proportional to the energy stored. It is possible to obtain a series of
readings by charging the capacitor to different values of p.d. and determining the rise in
temperature each time. We use this to show that energy E is proportional to V 2.
Page | 22
Ch 10.3: Clockwork models
Learning outcomes

The period of a harmonic oscillator is independent of its amplitude (isochronous).

The variation of displacement of a simple harmonic oscillator with time is sinusoidal,
having the general form s = A sin (2πft + φ) where φ is a phase angle. The
expressions s = A sin (2πft) and s = A cos (2πft) are often convenient.

The motion of a harmonic oscillator is governed by the differential equation
d2s
 a  (2f ) 2 s
2
dt
with
2f 2 

k
m
in the case of a mass m restrained by springs of spring constant k. That is, the
acceleration of a simple harmonic oscillator is proportional to its displacement and
always acts towards the equilibrium (zero displacement) position.
There are fixed phase relationships between the variations of displacement, force,
acceleration and velocity. In particular, there is a phase difference of π / 2 between
displacement and velocity, and between velocity and acceleration.
This section illustrates another major class of mathematical model, second order
differential equations, which describe the simple harmonic motion encountered in this
chapter. Oscillations occur in a great variety of phenomena, both natural and engineered
and those that approximate to simple harmonic motion lend themselves to the analysis
presented here.
Introducing oscillators
The opening group of activities is designed to give you a qualitative appreciation of a
range of oscillators. We begin with a circus of four distinct student activities. Additional
examples might be demonstrated and discussed if time permits.
Activity 200P Presentation 'The water pendulum'
Activity 210P Presentation 'Swinging bar or torsion pendulum'
Activity 220P Presentation 'Oscillating ball'
Key points:
◦ Many, but not all oscillators are isochronous.
◦ If the oscillations are isochronous the period does not depend on the amplitude.
◦ Oscillatory systems have an equilibrium position.
◦ The restoring force is always directed towards the equilibrium position.
◦ The time trace (displacement against time graph) is cyclic in nature but the amplitude
and period may change with time.
Activity 230P Presentation 'Mass oscillating between elastic barriers'
Page | 23
A number of real world oscillators are worth considering such as:

mountain bike suspension

tennis racket after striking a ball

chimney in a gusty wind

twanging a ruler on a desk

wind-induced oscillations in power cables

the tip of a fishing rod whilst fly-casting

oscillations in musical instruments

seiches
The next demonstration will be used as an introduction to modelling an oscillator.
Activity 240P Presentation 'Looking at an oscillator – carefully'
This demonstration provides an introduction to a type of oscillation known as
‘simple harmonic motion’.
Display Material 60O
OHT 'A language to describe oscillations'
Language to describe oscillations
Sinusoidal oscillation
+A
Phasor picture
s = A sin t
amplitude A
A
angle t
0
time t
–A
periodic time T
phase changes by 2
f turns per 2 radian
second
per turn
 = 2f radian per second
Periodic time T, frequency f, angular frequency :
f = 1/T unit of frequency Hz
 = 2f
Equation of sinusoidal oscillation:
s = A sin 2ft
s = A sin t
Phase difference /2
s = A sin 2ft
s = 0 when t = 0
sand falling from a swinging pendulum leaves
a trace of its motion on a moving track
s = A cos 2ft
s = A when t = 0
t =0
A sinusoidal oscillation has an amplitude A, periodic time T, frequency f =
1
and a definate phase
T
Notes:
Page | 24
The following exercise will help you to see how the equation s = A sin ωt works.
Let = 1 rad s-1 and A = 5 m. Remember that 1 radian = 180/π degrees.
(a) Use you calculator to calculate the missing data in the table below.
t/s
t/rad
0
0
t/
degrees
s /m
0
0
0.5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
2.4
(b) Use the axes below to plot a graph of s against t.
(c) (i) Deduce the period from the graph above. …………………………………………………
(ii) Now use the formula T = 2π/to calculate the period and compare the result you
obtain with the value obtained from the graph.
………………………………………………………………………………………………….
…………………………………………………………………………………………………
Page | 25
Display Material 70O
oscillator'
OHT 'Snapshots of the motion of a simple harmonic
Motion of a harmonic oscillator
displacement velocity
force
against time against time against time
large displacement to right
right
zero velocity
mass m
large force to left
left
small displacement to right
right
small velocity
to left
mass m
small force to left
left
right
large velocity
to left
mass m
zero net force
left
small displacement to left
right
small
velocity
mass m
left
small force to right
large displacement to left
right
zero velocity
mass m
large force to right
left
Everything abou t harm onic motion follows from the resto ring fo rce
bein g propo rtional to m inus the disp lacement
Page | 26
Modelling an oscillation
Having looked at a variety of oscillators and having begun to describe them, we now take
a closer look at the behaviour of a mass on a spring. We use the modelling exercise that
follows to develop a key idea behind harmonic motion, the link between acceleration (or
force) and displacement.
Simple harmonic motion (SHM)
The restoring force is directly proportional to the displacement from the equilibrium
position and is always directed towards it.
We can summarise this with the equation F = - ks.
With mass m and spring constant k the acceleration is given by
a=F=-ks
m m
We can write this as the second-order equation
d2s = – k s.
dt2
m
Firstly, we consider qualitatively the effect of changing the mass m and the spring constant
k in terms of the concepts of forces and inertia.
Activity 250S Software Based 'Oscillating freely'
using File 130L Launchable File 'Modelling springs and masses'
We can then follow this with Activity 260E Experiment 'Loaded spring oscillator'. In this
experiment you can establish experimentally the relationship between the frequency or
period of oscillation and the key factors of mass and spring constant. The results can be
compared with the model developed above.
Both should confirm that
T  m and T  1/k
and f  k and f  1/m
or more succinctly, T  (m /k) and f  (k/m).
Page | 27
For those wishing to see a more mathematical approach we can derive expressions for
period and frequency as follows, and will gain a deeper insight into the phase relationships
between displacement, velocity and acceleration.
If s = A cos 2ft where A is the amplitude,
v = ds = -2fA sin 2ft
dt
and
a = dv = - (2f)2A cos 2ft = - (2f)2s.
dt
But we know that a = d2s = - k s.
dt2
m
Hence, (2f)2 = k and f = 1 (k/m). It follows that T = 2 (m/k).
m
2
The equations for s, v and a confirm the phase difference of /2 between s, v and a.
Use the space below to sketch graphs of s, v and a against time t.
Note that the maximum velocity and acceleration are as follows:
vmax =
2fA =
amax =
(2f)2A =
ωA
ωA
Page | 28
Display Material 100O
OHT 'Graphs of simple harmonic motion'
Force, acceleration, velocity and displacem ent
Phase differences
Time traces
varies with time like:
dis plac em ent s
/2 = 90
/2 = 90
 = 180
cos 2ft
... the velocity is the rate of change
of displacement...
–sin 2ft
... the acceleration is the rate of
change of velocity...
–cos 2ft
...and the acceleration tracks the force
exactly...
–cos 2ft
ve locity v
acc eleration = F/m
same thing
zero
If this is how the displacement varies
with time...
force F = –k s
dis plac em ent s
... the force is exactly opposite to
the displacement...
cos 2ft
Graphs of displacement, velocity, acceleration and force against tim e have sim ilar shapes but
differ in phase
Page | 29
The following Display Materials introduce the step-by-step method of predicting time
traces.
Display Material 80O
OHT 'Step by step through the dynamics'
Dynamics of a harmonic oscillator
How the graph starts
How the graph continues
zero initial velocity would stay
velocity
zero if no force
force changes
velocity
force of springs accelerates mass towards centre,
but less and less as the mass nears the centre
change of velocity
decreases as
force decreases
new velocity
= initial velocity
+ change of
velocity
trace curves
inwards here
because of
inwards
change of
velocity
t
0
0
time t
trace straight
here because no
change of
velocity
no force at centre:
no change of velocity
time t
Constructing the graph
because of springs:
force F = –ks
t
change in displacement = v  t
t
if no force, same velocity
and same change in
displacement
plus
extra change in
displacement from
change of velocity due
to force
extra displacement = –(k/m) s (t) 2
acceleration = F/m
acceleration = –(k/m) s
change of velocity  v
= acceleration  t
v = –(k/m) s t
extra displacement
= v t
Health warning ! This simple (Euler) method h as a flaw. It always changes the
displacement by too much at each step . This means that the oscillator seems
to gain energy!
Here we show how to assemble descriptions of the dynamics of the simple harmonic
oscillator to predict its motion.
Page | 30
Display Material 90O
OHT 'Rates of change'
Changing rates of change
dt
slope = rate of change of displacement
= velocity v
ds = v dt
s
v=
rate of change = rate of change
of slope
of velocity
ds
dt
= acceleration a
t
new slope = new rate of change of
displacement
dt
ds = v dt
= new velocity (v + dv)
dt
s
a=
new ds = (v + dv) dt
dv
dt
ds = (v + dv) dt
dv = a dt
t
dt
v dt
change in ds = d(ds) = dv dt
dt
= a dt 2
v dt
s
dv dt
d ds d2 s
= 2=a
dt dt
dt
( )
change in ds = d(ds) = dv dt = a dt2
t
The first derivative ds/d t says how steeply the graph slopes
The second derivative d 2s/dt 2 says how rapidly the slope changes
This is a graphical representation of rates of change, and how they change with time.
You now have a formula for drawing a graph of displacement against time for a simple
harmonic oscillator so you may be asked to try out this technique at this stage. Remember
that with all step-by-step methods there are errors introduced at each stage.
Page | 31
You should now work through the following two software activities in class.
Activity 270S Software Based 'Build your own simple harmonic oscillator'
Step-by-step calculations allow you to predict the future, as explored in chapter 9. Here
you put your knowledge of these steps to good use, building a model of an oscillator.
Activity 280S Software Based 'Step by step though an oscillation'
using File 140L Launchable File 'Model showing graphical steps to SHM'
This model is live, so you have a real chance to understand the process, seeing how
changes in time step, spring constant and mass affect all calculated steps, and how the
initial displacement affects the subsequent steps.
Page | 32
Ch 10.4: Resonating
Learning outcomes

The energy stored in a stretched spring, at extension x if the stretching force F = kx
is ½kx2

The energy stored in a mechanical oscillator is the sum of its potential energy ½kx2
and its kinetic energy ½mv2. At maximum extension and zero velocity, all the
energy is stored in potential energy of the spring; at zero extension and maximum
velocity all the energy is stored as kinetic energy of the moving mass.

An oscillator driven by a sinusoidally varying force responds strongly and has a
large amplitude if the force varies at or close to its natural frequency. This is
resonance.

The range of frequencies over which a resonator responds with large amplitude
varies with the amount of damping. The width of the resonant response curve
increases as the damping increases.
Energy in a mass spring oscillator
We start by looking at the energy exchanges in a mass-spring system.
Activity 370S Software Based 'Energy in oscillators'
using File 170L Launchable File 'Models looking at energy in simple harmonic oscillators'
An oscillating mass between springs can be described in terms of the energy exchanges
taking place – between elastic energy stored in the springs and kinetic energy carried by
the moving mass. The exchange happens because of the action of the force; energy
transferred is force × distance.
Notes:
Page | 33
Display Material 120O
OHT 'Elastic energy'
Energy stored in a spring
area below graph
= sum of (force 
change in displacement)
extra area
F 1 x
F1
total area
1
Fx
2
0
x
0
unstretched
extension x
force F 1
work F1 x
no force
larger force
Energy supplied
small change x
energy supplied = F x
F=0
x=0
F = kx
x
stretched to extension x by force F:
1
energy supplied = 2 Fx
spring obeys
Hooke’s law: F = kx
energy stored in stretched
spring = 12 kx2
1
Energy stored in a stretched spring is 2 kx 2
The relationship between force to extend a spring, and extension, determines the energy
stored.
Page | 34
Display Material 130O
OHT 'Energy flow in an oscillator'
Energy flow in an oscillator
displacement
potential energy
= 12 ks2
0
s = A sin 2ft
time
energy in stretched spring
potential energy
0
PE =
1
2
kA2 sin22ft
time
mass and
vmax
spring
oscillate
A
vmax
A
vmax
energy carried by moving mass
kinetic energy
0
KE =
1
2
2
mvmax
cos22ft
time
velocity
kinetic energy
= 12 mv2
v = vmax cos 2ft
0
vmax = 2fA
time
from spring to
moving mass
energy in
stretched spring
from spring to
moving mass
energy in
moving mass
from moving
mass to spring
from moving
mass to spring
The energy stored in an oscillator goes back and forth between stretched spring and moving
mass, between potential and kinetic energy
The energy sloshes back and forth between being stored in a spring, and carried by the
mass.
Page | 35
Resonance
The video 'The Tacoma Narrows bridge collapse' provides an interesting introduction to
this topic.
We start the practical work with a variety of simple activities that will give some experience
of resonant effects and damping.
Activity 320E Experiment 'Book on a string'
Activity 330E Experiment 'Resonance of a milk bottle'
Activity 350E Experiment 'Resonance of a mass on a spring'
Although we are concerned only with a qualitative understanding of the phenomenon of
resonance you will gain much by attempting to produce frequency/amplitude graphs using
Activity 340E Experiment 'Resonance of a hacksaw blade' or any suitable variant.
When the driving frequency matches the natural frequency of an oscillator the amplitude of
oscillation can rise dramatically. This is resonance. This experiment gets you to measure
how the amplitude of an oscillator changes with the frequency of the driver.
If time is short, the class may be divided so that half use a damped oscillator and half an
undamped one. It is important to measure the natural frequency so that the frequency at
maximum amplitude can be compared with it.
Display Material 140O
OHT 'Resonance'
Resonant re sponse
Example: ions in oscillating electric field
Oscillator driven by oscillating driver
electric
field
low damping:
large maximum response
sharp resonance peak
+
–
+
–
ions in a crystal
resonate and
absorb energy
m ore damping:
smaller maximum response
broader resonance peak
10
10
5
5
narrow range
at 12 peak
response
1
0
wider range
at 12 peak
response
1
0
0
0.5
1.5
1
frequency/natural frequency
2.0
0
0.5
1
1.5
frequency/natural frequency
2.0
Resonant response is at maximum when the frequency of a driver is equal to the natural frequency
of an oscillator
The amplitude of the oscillator is maximum when the frequency of the
varying force matches the natural frequency of the oscillator.
You should be aware of the effect of damping. The width of the resonance
curve increases as the damping increases.
Page | 36
Having seen a few examples of resonance and produced sketch graphs of amplitude and
frequency it will be worthwhile to model what has been observed using the same
techniques met earlier in the chapter. The concepts of damping, driving force and energy
changes are crucial here and all brought out clearly in the activity.
Activity 360S
Software Based 'Modelling resonance'
using File 180L
Launchable File 'A model to look at resonance'
Here you model a complex situation. A simple harmonic oscillator is subject to two other
forces – a periodic driving force and a drag force, depending on velocity. The model allows
you to simulate experiments done in the laboratory, but perhaps more importantly, to see
how such a situation can be modelled, building the model out of many well understood
fragments.
The work on resonance can be concluded by considering the readings and discussing the
part resonance plays in the modern world.
Page | 37
Questions and activities additional to those listed in the Student Notes
Section
Essential
Optional
10.1
Read A2 text pp 1-7
Qu 1-7 A2 text p 8
Question 20S Short Answer
'Randomness and half-life'
Question 30S Short Answer 'Decay in
theory and practice'
Question 40S Short Answer 'Model
growth and sample decay'
Activity 70S
Software Based 'Models of
radioactive decay series'
using File 60L Launchable File 'Decay chains in
WorldMaker'
Activity 80S
Software Based 'Models of
bubble decays in foam'
using File 80L Launchable File 'Foam models'
Reading 10T
relic or fake?'
Text to Read 'The Turin shroud –
Question 5C
Comprehension 'First
steps in mathematical modelling'
Question 10C Comprehension 'Disposal
of radioactive waste'
10.2
10.3
Read A2 text pp 9-13
Qu 1-6 A2 text p 14
Question 90S Short Answer 'Radioactive decay
with exponentials'
Question 50S Short Answer 'Short
questions on charging capacitors'
Question 60S Short Answer 'Charging
capacitors'
Question 65S Short Answer 'Separating
charge'
Question 70S Short Answer 'Discharge
and time constants'
Question 80S Short Answer
'Discharging a capacitor'
Question 110S Short Answer 'Energy
stored in capacitors'
Question 120S Short Answer 'Energy to
and from capacitors'
Question 140S Short Answer 'Capacitors
with the exponential equation'
Activity 160S
to e'
Activity 170S
and powers'
Software Based 'Approximations
Software Based 'Exponentials
Activity 120D Demonstration 'Charging a
capacitor at constant current'
Reading 20T Text to read ‘ Modelling conquers
the Atlantic - eventually
Question 130D Data Handling 'Discharge
of high-value capacitors'
Read A2 text pp 15-21
Qu 1-6 A2 text p 22
Question 200S Short Answer 'Solving the
harmonic oscillator equation'
Question 150S Short Answer 'Revisiting
motion graphs'
Question 160S Short Answer 'Oscillators'
Question 170S Short Answer 'Energy
and pendulums'
Question 190S Short Answer 'Harmonic
oscillators'
Activity 290S Software Based 'Slopes and
models' using File 150L Launchable File 'Slopes
and notation'
Activity 300S Software Based 'Making links
with mathematics' using File 160L Launchable
File 'Models to explore connections between
representations'
Activity 310E Experiment 'The period of a
pendulum is not constant'
Activity 370E Experiment 'Oscillation: Trial by
video'
Display Material 110O OHT ‘Comparing models’
Page | 38
10.4
Read pp 23-27
Qu 1-6 A2 text p 28
Question 230X Exposition–Explanation 'Energy
in an oscillator: With calculus'
Question 220S Short Answer 'Bungee
jumping'
Question 240S Short Answer 'Oscillator
energy and resonance'
Question 250S Short Answer 'Resonance
in car suspension systems'
Activity 360E Experiment ‘Finding a resonant
frequency accurately’
Activity 390E Experiment ‘Modelling chaos’
Question 210D Data Handling 'Energy in
a simple oscillator'
Summary
File 190L Launchable file ‘Chaotic models’
Reading 30T
collapse'
Reading 40T
the evidence'
'The Tacoma Narrows bridge
'Tacoma Narrows: Re-evaluating
Qu 1-6 A2 text p 30
These notes draw almost exclusively on the resources to be found in Advancing Physics A2 Student’s Book and CD-ROM published by
Institute of Physics Publishing in 2000 and 2008. They are intended to be used in conjunction with these resources and others not
specified.
John Mascall
The King’s School, Ely, Cambs
Page | 39