Science notes
Science notes
Predator–prey populations – snowshoe hares
and lynxes revisited
Ian Carter
9
A simple method for investigating
respiration in yeast
Eric Gilhooley
11
A cost-effective and environmentally
friendly method of titration
Mala Das Sharma
12
A model for explaining stationary waves,
final visit
J. C. E. Potter
15
Using role-play to model a mass spectrometer
Emily Perry
18
Obesity and the body mass index
John Rousseau
19
Making moment of inertia accessible to
students
Saouma BouJaoude 21
and Garine Santourian
Helpdesk
25
The SSR Writing Workshop, 2006
27
Predator–prey populations – snowshoe hares and lynxes revisited
Ian Carter
Many school textbooks and revision sites (for
example, Rockwood, 2006; BBC Bitesize revision
site) quote the Hudson Bay Company pelt data
on the cyclical nature of snowshoe hare and lynx
populations as evidence of a predator–prey relation-
ship, with the number of predators (lynxes) peaking
just after the prey (snowshoe hare) – see Figure 1.
The reasoning goes that the lynxes peak when more
food is available, but when none is left the number of
predators falls rapidly. Elton and Nicholson originally
Figure 1 Predator–prey relationship between lynxes and snowshoe hares as typically depicted in many
textbooks.
School Science Review, March 2007, 88(324)
9
Science notes
published this work in 1942. The data was not based
on systematic population surveys but taken from the
ledgers of the Hudson Bay Company’s pelt-buyers.
The conclusions were that these two populations
follow the Lotka-Volterra population model, with
the lynx controlling the snowshoe hare population.
In 1995, Krebs et al. undertook field experiments
and time-series analysis on this relationship. Krebs
experimented by using fertiliser on a number of 1 km2
boreal forest plots to promote plant growth and in
some plots predators were excluded. He concluded
that the hare cycle could only be understood as an
interaction involving the hare population, its food
supply, and a community of predators (not just the
lynx). Krebs concluded from his work that the hare
cycle is not driven primarily by plant–herbivore
interactions. Food limitation increased predation by
forcing hares to search more extensively for food,
making them less healthy, and by making them less
likely to escape predation. When the hare cycle is
in the down phase, predators (for example wolves,
lynxes, coyotes, great horned owls and goshawks)
turn to other food sources and very few examples
have been found of predator mortality being due to
starvation. The entire community of predators drives
the hare cycles, not just lynx. The data collected by
Krebs showed a population change for the snowshoe
hare from 174 km–2 in 1989 down to 3 km–2 in 1992.
The lynx population declined from 23 to 6 per km2.
Further investigations using radio collars showed that
the majority of lynx had moved away, with collars
found up to 800 km away in Alaska and British
Columbia. A twist to this story is that on Anticosti
Island in eastern Canada where there are no lynx the
hare cycle continues.
There is good evidence here of the acceptance
of a data set and explanation without the necessary
rigorous underpinning experimental work. Elton is
one of the great figures of the science of ecology
so acceptance by the teaching and textbookwriting community of his work has perhaps led to a
potentially simplistic interpretation which does not
reflect the situation in the wild accurately. Could this
small example be used by teachers as an example of
the role of evidence in the How science works section
of the new GCSE specifications (Roberts and Gott,
2006) and perhaps persuade textbook writers to look
for other examples of predator–prey models or at
least put some caveats in their texts?
References
BBC Bitesize revision site:
http://www.bbc.co.uk/schools/gcsebitesize/science
Elton, C. S. and Nicholson, M. (1942) The ten-year cycle in
numbers of lynx in Canada. Journal of Animal Ecology, 11,
215–244.
Krebs, C. J., Boutin, S., Boonstra, R., Sinclair, A. R. E.,
Smith, J. N. M., Dale, M. R. T., Martin, K. and Turkington,
R. (1995) Impact of food and predation on the snowshoe
hare cycle. Science, 269, 1112–1115.
Roberts, R. and Gott, R. (2006) The role of evidence in the
new KS4 National Curriculum for England and the AQA
specifications. School Science Review, 87(321), 29–39.
Rockwood, L. L. (2006) Introduction to population ecology.
Oxford: Blackwell.
Ian Carter is currently headmaster of Poole Grammar School. He was formerly head of biology and director
of science at Woodbridge School, Suffolk, and still has a keen interest in teaching ecology.
Email: [email protected]
10
School Science Review, March 2007, 88(324)
Science notes
A simple method for investigating respiration in yeast
Eric Gilhooley
There are a number of methods for investigating
respiration in yeast. These include conventional
respirometers, fermentation tubes, the collection of
carbon dioxide evolved, and the increase in volume
of a sample of dough (Clegg et al., 1996; Roberts
and King, 1987). Such methods can be difficult
and time consuming to set up, expensive in terms
of the amount of equipment required, or limited in
accuracy in terms of provision of quantitative data.
I developed the following idea when looking for a
simpler procedure that could be easily assembled
and manipulated.
The assembled apparatus is shown in Figure 1.
A straight length of glass tubing is inserted into a
rubber bung so that when the bung is placed in a
large test-tube, the glass tube reaches almost to the
bottom of the test-tube and projects about 30 cm
vertically upwards from the bung. A short length of
plastic tube can be placed over the top end of the
glass tube for safety. When a suspension of yeast is
placed in the test-tube and the bung inserted, any
carbon dioxide evolved collects in the top of the
test-tube. The increase in pressure forces the yeast
suspension up the glass tube. The rate of movement
of the yeast suspension can be measured over 1- or
2-minute intervals.
If the yeast suspension reaches the top of the glass
tube the bung can be removed to release the pressure
and recording of results restarted. Alternatively, a
valve could be inserted into the bung to allow the
release of carbon dioxide produced.
I have used the apparatus with key stage 4 (14 – 16
year-olds) and sixth-form groups (16–18 year-olds)
to investigate how factors such as temperature,
glucose concentration and alcohol concentration
influence the rate of respiration in yeast.
Use of the apparatus introduces two errors in the
experimental approach:
● reduction in volume of yeast suspension in the
test-tube as liquid rises up the tube;
● the effect of temperature on the volume of carbon
dioxide evolved.
The first error can be eliminated by timing how
long it takes for the level of yeast to rise between
two marks on the glass tube. The second error could
2-a56*c 5a'&6< 6*2
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-a4(& 6&56 67b&
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Figure 1
be determined and allowed for by calculation. In
practice, the apparatus gives very good results. The
flaws in the procedure allow more-able students
to access higher marks for the evaluation of their
coursework.
There can be problems with viability of yeast
when a suspension is prepared. These can usually
be overcome by preparing the yeast suspension
using a suitable nutrient mixture and incubating the
suspension at body temperature for about 60 minutes
before it is needed. A suitable nutrient mixture (Clegg
et al., 1996) is listed below. The amounts given are
required to prepare 1 dm3 of nutrient medium (the
appropriate CLEAPSS Hazcards should be referred
to before the solution is prepared):
glucose
NH4Cl
KH2PO4
KCl
MgSO4
yeast extract
20.0 g
2.7 g
0.7 g
0.7 g
1.2 g
2.0 g
Different technicians and teachers will have their
own preferences for the amount of yeast to use when
preparing a suspension, but 5 g of dried bakers’ yeast
per dm3 of liquid usually gives good results.
References
Clegg, C. J. with Mackean, D. G., Openshaw, P. H. and
Reynolds, R. C. (1996) Advanced biology principles and
practices. Study guide. London: John Murray.
Hazcards (various dates) CLEAPSS. www.cleapss.org.uk
Roberts, M. B. V. and King, T. J. (1987) Biology a functional
approach. Students’ manual. 2nd edn. London: Nelson.
Eric Gilhooley is head of science at St Robert of Newminster School and Sixth Form College, Washington,
Tyne and Wear NE38 8AF. Email: [email protected]
School Science Review, March 2007, 88(324)
11
Science notes
A cost-effective and environmentally friendly method of titration
Mala Das Sharma
Owing to the hazardous effects and cost of many
chemicals, experiments are now being conducted so
that the use of chemicals can be kept to a minimum.
I am suggesting a microtitration method in which
the apparatus used for conventional titration is
replaced by weighing bottles, disposable syringes
and a standard digital balance. The method is
applied to a redox titration where estimation of Fe2+
by dichromate(VI) is carried out and the results are
comparable with those obtained by the conventional
method (Jeffery et al., 1991). The method described is
low risk, economical and environmentally friendly.
Method
The change in this method is the measurement
of mass (instead of volume), as standard singlepan digital balances capable of yielding quick and
accurate mass measurement in a short span of time
are now available.
The density ρ2 of potassium dichromate(VI)
(TOXIC) solution (titrant) is first determined with
the help of a specific gravity bottle (relative
density bottle). A known small volume (V1) of the
ammonium iron(II) sulfate (also known as Mohr salt)
solution of unknown concentration, which contains
Fe2+, is transferred into a Pyrex glass weighing bottle
by a syringe attached with needle. Two syringes
of capacities 1 and 2 cm3 are used for V1 < 1 cm3
and ≥ 1 cm3, respectively. Before transferring the
solution, care must be taken to expel any air bubble
from the syringe by taking solution in a little in
excess of V1. A previously prepared mixture of
H2SO4 (3 mol dm–3) (CORROSIVE), H3PO4 (16 mol
dm–3) (CORROSIVE) and water in the volume ratio of
20:3:40 is taken out by another syringe of capacity
5 cm3 (~3.15 times V1) and added to the ammonium
iron(II) sulfate solution. One drop of diphenylamine
sulfonate indicator is then added and the resultant
mass is recorded (W1). Using a dropper, potassium
dichromate(VI) solution is added dropwise, with
continuous shaking, until the endpoint is reached
(colourless to blue-violet) and the mass is noted
down (W1*). The difference between W1* and W1
will give the mass of titrant added (W2 = W1* – W1).
The entire process is repeated for different volumes
of ammonium iron(II) sulfate solution ranging from
0.2 to 1.2 cm3.
12
School Science Review, March 2007, 88(324)
Results and discussion
The results of the experiment are presented in
Table 1. It can be noted that although mass of
dichromate(VI) solution is determined, its volume at
endpoint V2 is calculated from the measured mass
W2 and density ρ2 (where V2 = W2/ρ2) and there is no
conceptual deviation in the calculation of strength
of the ammonium iron(II) sulfate solution by this
new method. The concentration of the unknown
ammonium iron(II) sulfate solution M1 can be
calculated from M1 = (M2 × V2 × n1)/(n2 × V1), where
M2 is the concentration of dichromate(VI) solution,
V1 is the volume of ammonium iron(II) sulfate
solution and n1, n2, are the numbers of moles of
Fe2+ and Cr2O72– (see also Table 1). The value of M1
calculated for each observation is given in Table 1.
In addition to the estimation of M1 independently
for each observation as described above, a graphical
method can also be adopted for the calculation
of M1 when the number of observations is three
or more. Figure 1 shows the plot of volume of
ammonium iron(II) sulfate solution against volume
of dichromate(VI) solution, which is a straight line
passing through the origin with a high correlation
coefficient (R2 = 0.998). From the slope of the line as
well as the concentration of known dichromate(VI)
solution (M2), the concentration of the ammonium
iron(II) sulfate solution M1 is determined. Excellent
agreement can be observed for estimated values of
M1 by the present microtitration method and the
conventional method of titration (see Table 1).
Environmental considerations
In the conventional method, after performing the
titration, the remaining potassium dichromate(VI)
solution is normally poured down the sink.
Hexavalent chromium remains in soluble form
during migration and may contaminate groundwater.
Owing to its high oxidising potential, Cr6+ can easily
penetrate the biological membrane, causing cancer
risk and kidney damage. The new method described
here will thus be beneficial in terms of handling
substantially reduced quantities of hazardous
dichromate(VI) solution. Moreover, it is suggested
that the small quantity of dichromate(VI) solution
left behind after titration should be reduced to Cr3+
before pouring into the sink.
Science notes
Table 1
Results of titration between ammonium iron(II) sulfate solution − Fe2+ (unknown) and
potassium dichromate(VI) solution (TOXIC), having concentration M2 = 0.016 mol dm–3 and
density ρ2 = 1.0049 g cm–3.
Indicator: diphenylamine sulfonate
Endpoint: colourless to blue-violet
Equation: Cr2O72− + 6Fe2+ + 14H+ → 2Cr3+ + 6Fe3+ + 7H2O
1Cr2O72− ≡ 6Fe2+
No. of observation
1
2
3
4
5
6
Volume of Fe2+ solution (V1) cm3
0.2
0.4
0.6
0.8
1.0
1.2
Mass of Fe2+ solution + other reagents
+ indicator (W1) g
0.892
1.813
2.704
3.640
4.427
5.340
W1 + mass of Cr2O72− solution till endpoint (W1* ) g
1.132
2.241
3.370
4.483
5.489
6.630
Mass of Cr2O72− solution required for titration
(W2 = W1* – W1) g
0.240
0.428
0.666
0.843
1.062
1.290
Volume of Cr2O72− solution required for titration
(V2 = W2/ρ2) cm3
0.239
0.426
0.663
0.839
1.057
1.284
Molarity M1 of unknown Fe2+ solution
M1 = (M2 × V2 × n1)/(n2 × V1) mol dm–3
0.1147
0.1022 0.1061 0.1007 0.1015 0.1027
From Figure 1 the slope of the line m = 1.0671 = V2/V1.
Therefore M1 = 1.0671 × 0.016 × 6 = 0.1024 mol dm–3
Using the conventional titration method where 10 cm3 ammonium iron(II) sulfate solution needed 10.6 cm3
of dichromate(VI) solution to yield the end point, M1 = (0.016 × 10.6 × 6)/10 = 0.1018 mol dm–3
Volume of titrant V2 cm3
y = 1.0671x
1.6
Figure 1 Plot of volume of
ammonium iron(II) sulfate
solution (V1) against potassium
dichromate(VI) solution (TOXIC)
(V2). The slope of the line
m = V2/V1 = 1.0671 is used for
calculating the concentration
of the ammonium iron(II) sulfate
solution (see Table 1).
R2 = 0.998
1.2
0.8
0.4
0
0
0.2
0.4
0.6
0.8
1.0
1.2
Volume of analyte solution V1 cm3
School Science Review, March 2007, 88(324)
13
Science notes
Sources of error and precautions needed
1 In the method described here, division by
density of the mass of dichromate(VI) solution
required to reach the endpoint of a titration,
is used to calculate volume (V2 = W2/ρ2). It is
therefore necessary to determine the density
of dichromate(VI) solution accurately. A highcapacity specific gravity (relative density) bottle
(~ 25 cm3 or more) should be used for this
purpose.
2 If a single observation (reading) is taken, the
microtitration method may not be suitable for
extremely small volumes of ammonium iron(II)
sulfate solution (~ 0.2 cm3). Table 1 shows greater
deviation for estimated values of M1 compared
with those obtained for higher volumes ranging
from 0.4 to 1.2 cm3, as well as the conventional
titration.
Advantages
The method presented here has several advantages:
1 The amounts of reagents used are small and hence
the method is cost-effective and the amount of
waste is minimal.
2 The experiment can be performed within a short
time once the digital balance is stabilised.
3 Use of burettes and pipettes is avoided, reducing
replacement costs of these easily broken items,
and storage space required in the laboratory is
reduced.
4 The accuracy of the result is unaffected for
V1 ≥ 0.4 cm3.
Disadvantage
The only disadvantage of the proposed method is the
relatively high initial cost of a digital balance (many
times greater than a burette and pipette). However,
once it is available, many students performing
individual experiments can share one balance
because readings are displayed quickly.
Conclusion
The microtitration method described here yields
excellent results and has other inherent advantages.
It could be useful for teachers/demonstrators of
advanced level in school and for undergraduate
classes.
Reference
Jeffery, G. H., Bassett, J., Mendham, J. and Denney, R. C.
(1991) Vogel’s textbook of quantitative chemical analysis
(5th edn, revised), Chapter 10, pp. 257. Harlow: Longman.
Mala Das Sharma, St Pious X Degree and PG College for Women, Snehapuri Colony, Nacharam,
Hyderabad-500076, India. Email: [email protected]
14
School Science Review, March 2007, 88(324)
Science notes
A model for explaining stationary waves, final visit
J. C. E. Potter
At the end of my science note ‘A model for explaining
stationary waves revisited’in the December 2005 SSR
(Potter, 2005) I said ‘a motor-driven model would be
something!’ I have now managed to do this and it
was exhibited at the Bournemouth Natural Science
Society open day on 2 September 2006 (Figures 1
and 2). The model ran non-stop all day.
A microwave oven motor was chosen because of
its size, power and slow rate of rotation. A microwave
oven was obtained from the local council dump, on
payment of two pounds. I checked that the motor ran
before dismantling the oven to remove the motor
(oven subsequently returned to the dump). I rang
local venetian blind suppliers and found one that had
the blank aluminium strip by the roll (60 p a metre).
The wave was traced on it using a cardboard template,
8 cm wavelength, then cut by small tin snips and the
edges smoothed with fine emery paper. The motor
was attached to the underside of some chipboard
shelving by screws through its fixing flanges (see
Figure 3), with the drive shaft protruding clear of its
hole. A piece of plywood covers the mains wiring
connectors. Use was made of the wooden revolving
bases, cut circular, of two CD holders. The bases had
a useful ball-bearing fitment, unnecessary for my CD
storage; the holders are more stable without them.
One base was used for the drive and one for the idler.
Instead of using jam jars to support the aluminium
loop, as in the earlier model, plastic cups were used
since they had sloping sides that prevented the loop
Figure 1 Visitors to the exhibition try out the model.
School Science Review, March 2007, 88(324)
15
Science notes
Figure 2 Side view of the completed model showing the waveband in front of carbon fibre guide rods.
from ascending when running. Two wide rubber
bands were put over the drive cup to prevent the
aluminium loop from slipping. The drive cup was
screwed to the driven base with three screws. The
continuous wave loop was joined using two small
nuts and bolts. The idler base, central cup fitted, was
then positioned so the loop was taut and then fixed
by a central screw, allowing free rotation. How the
Figure 3 The microwave motor used.
16
School Science Review, March 2007, 88(324)
drive shaft is attached to the drive base depends on
the method used in the oven. The one I used had a
square-ended plastic extension that I let into the drive
base (drill and square with a chisel) and fixed with a
strong adhesive (Araldite Super-Metal). The beads,
etc., were as for the hand-driven model described in
the December 2005 science note. I might have tried
cutting slots in two Perspex strips as guides for the
oscillating beads instead of rods, but I don’t have a
band saw. The chipboard shelving base was 9 inches
by 36. A batten 8 inches by 1 by 3/4 was screwed
underneath the base at each end to give the motor
ground clearance (2 inches in to give finger room
for lifting the model). The aluminium strip was 5
cm wide and the wave amplitude a quarter of the
width. For the exhibition, a plane mirror was placed
behind the model. I think the model was well worth
the effort (done in odd moments). Of course, its
manufacture could be farmed out to a class, Adam
Smith fashion.
Guide rollers (e.g. flame-polished glass tubing,
cut biro or ignition tubes) were slipped over the end
carbon-fibre uprights to ensure the loop cleared the
rest (Figure 4).
The two pieces of white plastic fascia are
cosmetic, as is the chipboard-base facing strip.
If the ball-bearing bases are unavailable I suggest
using a DVD packaging drum. The Sony 25 DVD
drum is best, giving less play and a bigger flange
(Figure 5). Always check satisfactory rotation by
holding the central pillar pointing down and placing
the top of the outer case on it. Other drum makes
may stick or have too much play and not enough
flange. Using scissors, cut down the drum side to
about eight millimetres from its base floor. Cut out
the inner central ring. Use the blank packing disc,
or an old DVD, to draw a circle on a piece of wood
about a centimetre thick and cut out the disc. The
wood disc goes inside the cut-down drum and is
Science notes
Figure 4 A plan view showing the mains switch and end-clearance rollers.
5). Or the base underside can be packed with three
unwanted DVDs or CDs to prevent distortion and
may then be drilled and screwed anywhere. The cutdown drum with wood insert should rotate freely in
it. Attach the motor drive shaft as before. Screw the
plastic cup centrally to the wood disc. Construct an
idler disc in the same way. A little thin oil will help
to reduce contact friction and wear.
Figure 5 Alternative revolving beaker base
constructed from Sony DVD drum pack.
glued to the drum floor. The top of the wood disc
should be smooth; if not, glue the packing disc to
the top so the waveband runs freely over its surface.
Remove the central disc stacking column with snips.
Make sure the central hole clears the motor drive
shaft. Screw it upside down centrally over the drive
shaft, without distorting the base. Use three roundhead screws, small enough to lie clear in the space
between the fixed and rotating discs (see Figure
Suppliers
Acrylic beads were obtained from Hobbycraft, www.
hobbycraft.co.uk, tel: 01202 582444. They order
from The Beadery, PO Box 169, Hertford SG14
2AS. The large round yellow beads 716A are 10 mm
in diameter, 30 per pack. The clear, faceted crystal
beads 712A 006 are 12 mm in diameter, 43 per pack.
The coloured faceted beads 712A 029 multi 12 mm,
38 per pack. All are 59p per pack. Carbon fibre rod, 2
mm diameter, costs £3–£4 for a 2-metre length, and
is available from kite or model shops. CD holders,
£10, from T. J. Hughes plc. www.tjhughes.co.uk
for store finder (49 stores). Blank aluminium strip
can be obtained from Christchurch Blinds Ltd, tel:
01202 478363.
References
Potter, J. C. E. (1998) Explaining stationary waves. School
Science Review, 79(289), 114–115.
Potter, J. C. E. (2005) A model for explaining stationary
waves revisited. School Science Review, 87(319), 47–50.
John Potter was head of science at Talbot Heath School, Bournemouth, from 1967 until reaching retiring
age in 1985.
School Science Review, March 2007, 88(324)
17
Science notes
Using role-play to model a mass spectrometer
Emily Perry
In an effort to involve my rather quiet AS chemistry
class (16/17-year-olds) more actively in their
learning, I decided to get them to act out the workings
of a mass spectrometer. There are relatively few
opportunities for practical work in the topic of
atomic structure, so this presented a good chance
for some active engagement in the lesson, especially
since I also asked the students to consider how to
model each part of the machine, given a couple of
props and a minimal amount of guidance.
We decided to use the layout shown in Figure 1,
with students taking turns to act as the sample passing
through the spectrometer. The sample students were
(sensitively!) chosen to represent atoms of different
masses. They followed the path shown on the
diagram, carrying a box of tennis balls to represent
electrons. Other students acted as the various parts
of the machine, indicated by the four stages on the
diagram, as follows.
Stage 1: At this first stage, the sample was
ionised by removal of an electron from the box.
We discussed as a class how best to model this
and decided against the more realistic but slightly
hazardous method of throwing other tennis balls
at the sample.
Stage 2: Here, the sample student was accelerated
by two students pushing him or her forwards.
Stage 3: At the third stage, the sample was
deflected by two more students. Again, we
discussed how this could be modelled and
decided to concentrate solely on the repulsion of
the sample from one side of the deflector. The key
teaching point, of course, is that heavier sample
ions require more force (a stronger magnetic
field) to deflect than lighter ions.
Stage 4: Finally, at Stage 4 the sample was
detected. The detectors noted the mass of the ion
(we did this qualitatively but they could have
weighing scales to measure its actual mass) and
its number of electrons (to give an idea of how to
calculate the mass/charge ratio).
Did it work?
The class clearly enjoyed this opportunity to
model a piece of equipment, and it produced much
laughter and some useful discussion of how the mass
spectrometer works, especially the formation and
deflection of the ions. We followed up the activity
with written questions on what happened at each
stage, and it was apparent that students had a good
understanding of the workings of the spectrometer.
It seemed that this was a particularly successful
activity because of the role students took in deciding
how best to model the process, rather than simply
being given a set of instructions.
With a bigger class, and therefore more sample
students, the idea could be extended to measure
the relative abundances of each ion, using students
of similar masses as ions of the same isotope. You
could also form M2+ ions (by removing two electrons
at Stage 1) and molecular samples (made of students
holding hands), to model more complex spectra.
Figure 1 Stages in modelling a mass spectrometer.
Emily Perry is a chemistry teacher at St Mary’s High School, Chesterfield.
18
School Science Review, March 2007, 88(324)
Science notes
Obesity and the body mass index
John Rousseau
I hope that biologists will bear with me in what
follows. It is not biology, but physics. I am not
attempting to bring physics to bear on what I know
is a complex and difficult issue, that of obesity itself,
but rather to question the validity of the index by
which it is recognised and measured, by means of
some simple physics. I had never heard of B (body
mass index) until my son-in-law mentioned that
Sir Steve Redgrave’s value of B made him obese!
Whether this is true or not I do not know, but it got
me thinking. The next time I came across B was
when I had to go for a medical check-up in the
course of which my own value of B was measured
by the district nurse; she had no opinions about it
– she just had to do it as part of her job. In spite of
the fact that I was not classed as obese I decided to
give the subject some careful thought.
Theory
A currently accepted measure of obesity is the socalled body mass index, which is defined to be:
M
h2
where B is the body mass index, M is the body mass
and h is the height of the individual. There are reasons
for thinking that this is a less than useful measure of
a person’s obesity. If M is measured in kilograms and
h in metres, then B will have units of kg m–2. If B
is to be useful over a range of heights and masses,
it should be dimension free, that is, it should be a
pure number or depend on intensive quantities (e.g.
density), which is clearly not the case here.
As a simple example of the problems with B,
consider the case of two individuals, one double the
height of the other, where the relative proportions of
each are exactly the same (that is, the height doubles,
and all the other body dimensions double, so that the
figures are exactly similar). Clearly, any sensible
measure of obesity would give these two individuals
the same value. However, if all the linear dimensions
double, the volume of the larger figure will be eight
times that of the smaller, and, assuming the same
density for each, the mass of the larger figure will
be eight times the mass of the smaller. So the value
of B in the case of the larger figure will be twice
that of the smaller, which is related to the fact that
the body mass index as so defined has non-intensive
dimensions, and so is scale-dependent. If significant
values of B have to be specified for each value of
body height because of this scale dependency, then it
B=
is hardly more useful than using the body mass alone
as a parameter of obesity.
Another criticism of this index is that it takes no
account of individual relative proportions. Consider
two people, both of the same height and mass. One is
a body builder with a large muscle mass and little fat,
and the other is obese with small muscle mass and a
lot of fat. The value of B for both these individuals
will be the same, which makes no sense.
An improved index
As a starting point for an improved index for obesity,
consider the following. As a first approximation,
assume that fat is deposited over the body in a uniform
way as a person gets fatter. Now the circumference
of the waistline (call this W) is proportional to the
mean diameter of the waistline, which in view of
the assumption above is proportional to the mean
diameter of the body as a whole, so the volume of
the body will be proportional to the height times the
square of W, that is:
V ∝ hW 2
Assuming that the density is roughly constant,
then the mass of the body is proportional to these
dimensions in the same way, that is:
M ∝ hW 2
This then means that:
W2 ∝
M
h
M
2
∝ 3
h
h
The left-hand side of this relationship is dimension
free (being a ratio of two lengths) and the right-hand
side has the dimensions of density, which is intensive
(i.e. independent of object dimensions in this case).
If all individuals were of the same proportions
then to a good approximation the values of M/h3
would all be the same regardless of height, as would
the values of (W/h)2. This is not the case of course
for the body mass (which is an increasing function of
height, even for individuals of the same proportions);
nor is it the case for the so-called body mass index
(which, like the mass, is an increasing function of
height even for individuals of the same proportions,
although not so rapidly increasing).
Using M/h3 as a parameter of obesity would
be possible, and preferable to using B. For a given
or:
(W )
School Science Review, March 2007, 88(324)
19
Science notes
height, a more obese individual would have a bigger
mass than a person of ideal proportions, leading to a
larger value of M/h3 than the ideal, but then so would
a person of bigger muscle mass (e.g. a body builder).
For this reason, it seems better to choose (W/h)2 or
even simpler W/h as a measure of obesity; a person’s
waistline is a better measure of extra fat than the
total body mass is.
Trial results
Data on a total of 56 men and 59 women were
obtained (see figures below). This is admittedly a
small sample; worse was the fact that nothing was
actually measured: people were encouraged to fill
in the details of their measurements on the basis of
what they thought their measurements were. Data
sheets were prepared and distributed to my local
K&06 2&4c&06a(& 8a4*a6*10 *0 V*) ab176 a8&4a(& 8a-7&
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,)2+
,)14
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=.+
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/+
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20
School Science Review, March 2007, 88(324)
Science notes
leisure centre, to the doctor and the dentist and to
shops and pubs in my area. The people involved
were very helpful once they understood what it was
all about. There might well be an element of wishful
thinking about some of the returns!
Conclusions
These are rather tentative in view of the nature of the
data collected, as described above. In the first place,
the least squares fit (the purple lines in the figures)
are without major trends as a function of height,
being effectively zero in the case of men, and small
in the case of women. For men, the average value
of W/h turns out to be 0.5 and for women 0.45 and
so one could say that the onset of obesity for men
occurs when the waistline significantly exceeds 50%
of height and for women 45%. No doubt better data
will refine this conclusion.
J. S. Rousseau is a retired physics lecturer and lives in Broughton Mills, Cumbria.
Email: [email protected]
Making moment of inertia accessible to students
Saouma BouJaoude and Garine Santourian
The approach used to teach moment of inertia in
many physics classes emphasises mathematical
application rather than conceptual understanding.
Typically, teachers start by presenting the formula
of moment of inertia for one point mass and then
move to more than one point masses located at
different positions. Finally, the formulas for a hoop,
disc, cylinder and sphere are given. Most of the time
students can easily calculate the moment of inertia
for a given solid by blindly plugging the value of
the mass (m) and the position from axis (r) in the
formula (I = mr2 for point mass). One problem with
such a method is that students do not understand the
physics behind the formula of moment of inertia.
Therefore, instead of using the mathematics as a tool
for solving physics problems, mathematics is hiding
the physics behind the formula.
Simply stating the definition of moment of
inertia, memorising its formula, and solving
numerical problems does not lead to conceptual
understanding. The concept is difficult and many
students may not be able to grasp it through this
approach. Such concerns are valid reasons to reverse
the traditional logic of teaching moment of inertia.
Thus came the idea of using an adapted version of
the learning cycle (Figure 1) in conjunction with a
discrepant demonstration to introduce the concept.
This adapted version starts by asking students to
predict the outcome of an activity, followed by the
steps of the traditional learning cycle: exploration,
concept introduction, and concept application
(Good, 1987).
To make the demonstration discrepant, two nontransparent identical cylinders of equal mass but
different mass distributions are used in this activity,
BWOJMQ>SGML
OQBAG@SGML
@ML@BOS
GLSQMAT@SGML
@ML@BOS
>OOJG@>SGML
Figure 1 An adapted learning cycle structure
(Good, 1987).
assuring that all variables are kept constant except
for the mass distribution (see Box).
When this activity was used in a physics class,
students were shown the two cylinders and asked to
predict which one would reach the end of an inclined
plane first and the reason for their predictions. The
students predicted that, if one of the cylinders was
heavier, it would roll faster and reach the end of
the inclined plane first. When the cylinders were
rolled, the students’ prediction was apparently
confirmed because one of the cylinders reached
the end first. However, to create disequilibrium in
students’ thinking, the teacher weighed the cylinders
and showed that their weights were identical. This
School Science Review, March 2007, 88(324)
21
Science notes
discrepancy between what was expected and what
actually occurred forced students to drop their initial
explanation and think of other factors such as mass
distribution inside.
However, thinking about mass distribution as a
possible explanation was not intuitive to students.
Consequently, they needed scaffolding to reach this
explanation. This was accomplished by using the
example of two rotating masses about an axis (see
Box). This example was used to bring students one
step closer to the correct answer without explicitly
giving them this answer. Based on their prior
experience, students were able to conclude that
masses closer to the axis of rotation have a higher
tendency to rotate. At this stage, students had
discovered an important fact, which is that change
in rotation depends on the position of a mass from
the axis. Furthermore, they could relate this example
to the demonstration and gradually explain the
discrepancy. This set the stage for introducing the
conceptual and meaningful definition of moment of
inertia. Finally, to reinforce the concept of moment
of inertia, students were provided with a variety of
examples from everyday life and required to explain
them using the concept (see Box).
BOX: The lesson plan
RACING CYLINDERS
Grade level: Grade 11 or 12 (ages 17 and 18 years approximately)
Subject: Physics
Purpose
Many students have the mistaken and over-generalised notion that if two cylinders are rolled down an
inclined plane the heavier cylinder reaches the end of the plane first. The purpose of this demonstration
is to address this misconception and help students realise that the acceleration of a cylinder depends on
its mass distribution and not on its mass. Racing cylinders provide excellent examples for introducing the
concept of moment of inertia.
Science content
● Moment of inertia is the tendency of a body rotating about an axis to resist change in this rotation
motion. The moment of inertia (I) of a particle of mass m and distance r from an axis of rotation (∆) is
given by:
I∆ = mr2
● The linear acceleration of objects rolling down an inclined plane depends on the mass distribution,
gravitational acceleration and the angle of inclination of the plane.
Objectives
At the end of this lesson, students will be able to:
● describe the moment of inertia of an object rotating about an axis;
● identify the factors affecting the linear acceleration of objects rolling down an inclined plane.
Materials
● Clothes pegs (these were used in this activity because they have identical masses, are readily
●
●
●
●
●
●
22
available, and are relatively cheap; if not available, they can be substituted with other items that serve
the same purpose)
Electronic balance
Hard cover books
Glue stick
Two identical non-transparent cylinders (e.g. cans)
Strips of thin metal sheets (the lengths of the metal sheets should be the same as that of the cylinders;
preferably, these metal sheets should be cut from cans similar to the cylinders)
Two smooth 85 x 20 cm wooden planks for the inclined plane
School Science Review, March 2007, 88(324)
Science notes
Preparation
Before class separate the two halves of all the clothes pegs. [Safety: Be careful when separating the
clothes pegs; the spring that holds the two halves may cause injury.] Glue the thin metal sheets and the
separated halves along the inside surface of one of the cylinders. Fill the entire volume of the second
cylinder with the separated halves of the clothes pegs. Make sure both cylinders have the same mass.
Put small marks on the cylinders to help you identify them. Now you have a hollow cylinder and a solid
cylinder of equal mass, diameter and height. Close both cylinders to make the mass distribution invisible to students. Before students enter the classroom, place the wooden planks on a pile of hardcover
books to form an inclined plane. Place some heavy books at the end of the inclined plane to catch rolling
objects. Smooth platforms give the best result because they minimise friction.
Instructional activities
1
Ask students:
a Is there any noticeable difference between the two cylinders?
b What do you think will happen if I roll both cylinders simultaneously from rest?
2
Roll both cylinders (Figure 2). Hold the hollow one on the side visible to the students so that they can
easily observe the solid cylinder overtaking the hollow one. Repeat the demonstration, if needed, so
students can pay careful attention and be convinced that one of the cylinders reaches the end before
the other.
)
)
mhollow = msolid
4
∆
rhollow = rsolid
4
hhollow = hsolid
∆
hollow cylinder of mass m
solid cylinder of mass m
Figure 2 Hollow and solid cylinders.
3
Weigh the two cylinders on the electronic balance to show that they weigh the same (if students
predict that the heavier one would reach the end first, the fact that both cylinders weigh the same will
cause them to rethink their explanation).
4
Since evidence from the activity suggests that weight is eliminated as a possible factor in the
explanation, ask the students to come up with other explanations. At this stage, do not reveal what
is inside the cylinders. It is up to the students to discover! Ask students to discuss the observations
and come up with an explanation for the discrepancy. Write students’ hypotheses on the board. Take
the hypotheses, one at a time, and ask them to think of methods for verifying their hypothesis.
5
If students do reach the correct
explanation, provide them with
the following example. Two
equal masses rotate about an
axis of rotation (∆). In Figure 3A,
the masses are further away
from the axis of rotation
compared to the position of
masses in Figure 3B. Ask:
Which one has the tendency
to rotate easily? (B) (Or Which
one makes greater number of
rotations per second? Why?)
a
∆
∆
A
B
Figure 3 Rotating masses.
School Science Review, March 2007, 88(324)
23
Science notes
b
c
What do you conclude from this example?
How is this example related to the demonstration?
Through their everyday life experiences, students should be able to discover that if the masses are closer
to the axis, then the number of turns per second is larger. Therefore, it is easier to rotate an object if its
mass is distributed closer to the axis of rotation. At this stage, students should be able to relate this
example to the demonstration. They can conclude that the mass of the solid cylinder is distributed closer
to the axis, whereas the same mass of the hollow one is concentrated away from the axis. Therefore, the
solid cylinder rolls faster because its mass distribution is even and closer to the axis.
6
Before revealing what is inside the cylinders, ask students to draw a model of the mass distribution
inside both cylinders. Let students draw the model on the board. Then emphasise that mass is
uniformly distributed in the solid cylinder, while it is concentrated on the periphery in the hollow one.
Reveal the mass distribution inside the cylinders.
7
Explain moment of inertia. Tell students that the solid cylinder is said to have less moment of inertia
than the hollow cylinder. Therefore, it has a higher tendency for rotation. The term moment of inertia
is defined as the quantity that measures resistance to changes in rotation. Moment of inertia depends
on the mass of the object and the distribution of its mass with respect to the axis of rotation. If the
position between an object’s mass and its axis of rotation is greater, then moment of inertia is larger
which means it tends to resist rotation.
8
To make sure students understand the principle of moment of inertia, ask them to explain again,
why the solid cylinder rolls faster. Have students summarise moment of inertia.
9
Application exercises. Now that students have seen examples of the moment of inertia of different
shapes, provide them with problems, preferably from everyday life, and ask them to explain them.
a In the lesson, the hollow and solid cylinders that were rolled down the inclined plane
were of equal mass. Do you think that mass affects the acceleration of the cylinders down the
inclined plane? Why or why not? How can you verify your answer?
b Which one is easier to rotate, a disc or a hoop? The disc, since its mass is evenly distributed with
respect to the axis of rotation:
Idisc < Ihoop since
c
1
2
mr 2 < mr 2 .
Which one will roll down a hill faster, a solid sphere or a solid cylinder? A solid sphere will always
beat all rotating objects because of its uniform even mass distribution around the axis:
Isphere < Icylinder since
d
2
5
mr 2 < 21 mr 2 (roll a solid ball and a solid cylinder to demonsttrate the result).
If the hollow cylinder was heavier than the solid one, which one would reach the end first? Some
students might think that because I depends on mass, then the heavier cylinder reaches the
ground faster. This is not true because acceleration does not depend on mass.
10 Secure a battery at the bottom of the hollow cylinder to make it heavier than the solid one. Roll the
two from rest. The solid cylinder will always beat the hollow one even if the latter is heavier.
Point out to students that acceleration does not depend on mass. Remind them of the free-fall example of
a feather and a stone landing together in a vacuum. In the absence of air resistance a feather and a stone
accelerate with the same gravitational acceleration, therefore reaching the ground together. The same
applies to a box moving from rest down an inclined plane (no rotation). The box accelerates because of
gravity and not mass. But in this case, one more factor affects acceleration, which is the angle of inclination of the plane. The greater the angle, the larger the acceleration will be. For objects rotating down an
inclined plane, the acceleration does not only depend on gravity and angle of inclination but also on how
mass is distributed about the axis (not the mass). Therefore, a solid cylinder rolls faster than a hollow one
irrespective of their masses.
24
School Science Review, March 2007, 88(324)
Science notes
Reference
Good, R. (1987) Artificial intelligence and science education. Journal of Research in Science Teaching, 24, 325–342.
Saouma BouJaoude is professor of science education and Garine Santourian is a physics teacher in the
Department of Education, American University of Beirut, Beirut, Lebanon.
Email: [email protected]
Helpdesk
Readers’ replies
The nature of matter
In December 2006 SSR, page 25, Jacqui asked:
What experimental evidence supports the claim that
thermal expansion is due to particles moving further
apart rather than expansion of particles?
I found Jacqui’s question (about particles and
expansion) fascinating, in that it challenges a
common assumption, and leads to consideration
at many levels. It would make an excellent test
for sixth-form students, or even ambitious 15–16
year-olds.
My first take was (accepting the usual textbook
model as true) to wonder why there seems to be
no easily demonstrated direct refutation of the
‘erroneous’ or ‘alternative’ concept, that is, that
particles expand on heating (as John Dalton himself
believed). The evidence is the thermal expansion of
solids and liquids, and, although this is now known
for most substances, it varies amongst them and can
be very small (e.g. Pyrex glass, or Invar metal).
Jacqui’s question alludes to the two different
explanations (ideas) for this bulk expansion:
(A) It is due to the expansion of each of the
particles.
(B) It is a consequence of the increased separation
of particles as they gain kinetic energy, and so
move faster and collide with greater force.
I suppose that the conflict between these two models
is resolved, not by the experimental refutation of
A, but by the success of B, or rather, the success of
kinetic molecular models as a family. This success
is most obvious in the case of gases, and it is easily
shown (e.g. in a glass syringe) that a mole of any
substance increases its volume on vaporisation, by
a factor of 1000 or so. This implies that gases have
an intermolecular separation of around 10 molecular
diameters, so any change in such a diameter, on
heating, would be negligible.
But, remember, we are not concerned with gases,
but with condensed matter. Well, the movement
of molecules in liquids can be demonstrated by
diffusion, or by Brownian motion, and it certainly
increases with temperature, so model B is supported.
Doesn’t this, then, disprove model A? I think not,
for the two are not mutually exclusive. Presumably
as, on heating, the molecules move apart they
do not leave little pockets of vacuum. We have to
reassess our concept of the ‘volume’ of a particle,
which is not like a ball-bearing, but maybe more
like a rather squashed thistle-down. And since, by
A-level, students will have to accept that rates of
molecular rotation and atomic vibration increase
with temperature, why not include the size of their
electron clouds?
As so often in science teaching, the problem
is knowing where to stop elaborating the model.
For 11–13 year-olds I would be content to show
molecular motion in liquids, for example by the
spread of colour around a ‘permanganate’ crystal
in a Petri dish of still water: and then to leave it
as obvious that the more agitated molecules must
spread apart, too. And to the 12-year-old my reply
might be: ‘Well done! Splendid idea. It could well be
a part of the reason.’
Nick Selley
Retired, Bath
Email [email protected]
I suggest that the topic is far too advanced. The
idea that matter can be divided into small particles
can be taught to young children, but there remains
School Science Review, March 2007, 88(324)
25