Module: Science Skills and Safety Time allocation: 8 hours

About This Presentation

Title:

Module: Science Skills and Safety Time allocation: 8 hours

Description:

IJSO Training Course Phase III Module: Science Skills and Safety Time allocation: 8 hours * 2. Introduction: If the experiment is designed to verify a law, state the ... – PowerPoint PPT presentation

Number of Views:178

Avg rating:3.0/5.0

Slides: 89

Provided by: phy73

Category:

more less

Transcript and Presenter's Notes

Title: Module: Science Skills and Safety Time allocation: 8 hours

1
Module Science Skills and Safety Time
allocation 8 hours
IJSO Training Course Phase III
2
Objectives

Describe, distinguish between random
uncertainties and systematic errors.
Define and apply the concept of significant
figures.
Identify and determine the uncertainties in
results calculated from quantities and in a
straight-line graph.
Introduce general procedures of writing
experimental reports
Laboratory safety and rules.

3
1. Significant Digits

Suppose you want to find the volume of a lead
cube. You could measure the length l of the side
of a lead cube to be 1.76 cm and the volume 13
from your calculator reads 5.451776. The
measurement 1.76 cm was to three significant
figures so the answer can only be three
significant figures. So that the volume 5.45
cm3. The following rules are applied generally

All non-zero digits are significant. (e.g., 22.2
has 3 sf)
All zeros between two non-zero digits are
significant. (e.g., 1007 has 4 sf)
For numbers less than one, zeros directly after
the decimal point are not significant. (e.g.,
0.0024 has 2 sf)
A zero to the right of a decimal and following a
non-zero digit is significant. (e.g., 0.0500 has
3 sf)

All other zeros are not significant. (e.g., 500
has 1 sf, 17000 has 2 sf)
When multiplying and dividing a series of
measurements, the number of significant figures
in the answer should be equal to the least number
of significant figures in any data of the series.

For example, if you multiply 3.22 cm by 12.34 cm
by 1.8 cm to find the volume of a piece of wood,
you get an initial answer 71.52264 cm3 from your
calculator. However, the least significant
measurement is 1.8 cm with 2 sf. Therefore, the
correct answer is only 72 cm3.
When adding and subtracting a series of
measurements, the answer has decimal places with
the least accurate place value in the series of
measurements.

For example, what is your answer by adding 24.2 g
and 0.51 g and 7.134 g? You get an initial answer
31.844 g from your calculator. However, the least
accurate place measurement is 24.2g with only one
decimal point. So the answer is 31.8 g.

Exercises
1. How many significant figures are indicated by
each of the following?
(a). 1247 (b). 1007 (c). 0.0345
(d). 2.20 x 107 (e). 62.00 (f). 0.00025
(g). 0.00250 (h). sin 45.2o
(i). tan-10.24 (j). 3.20 x 10-16
(a). 4 (b). 4 (c). 3 (d). 3
(e). 4
(f). 2 (g). 3 (h). 3 (i). 2
(j). 3

2. (a) Add the following lengths of 3.15 mm and
7.32 cm and 19.2 m.
(b) A rectangular box has lengths of 2.34 cm,
90.66 cm and 3.7 cm. Calculate the volume of the
box cm3.
(a) 0.00315 0.0732 19.3 19.27635. Thus the
answer is 19.3 m.
(b) 2.34 x 90.66 x 3.7 784.93428. Thus the
answer is 780cm3.

10
2. Making Measurements

A measurement should always be regarded as an
estimate. The precision of the final result of an
experiment cannot be better than the precision of
the measurements made during the experiment, so
the aim of the experimenter should be to make the
estimates as good as possible.

There are many factors which contribute to the
accuracy of a measurement. Perhaps the most
obvious of these is the level of attention paid
by the person making the measurements a careless
experimenter gets bad results! However, if the
experiment is well designed, one careless
measurement will usually be obvious and can
therefore be ignored in the final analysis.

Systematic Errors
If a voltmeter is not connected to anything else
it should, of course, read zero. If it does not,
the "zero error" is said to be a systematic
error all the readings of this meter are too
high or too low. The same problem can occur with
stop-watches, thermometers etc.

Even if the instrument can not easily be reset to
zero, we can usually take the zero error into
account by simply adding it to or subtracting it
from all the readings. (However, other types of
systematic error might be less easy to deal
with.)
For this reason, note that a precise reading is
not necessarily an accurate reading. A precise
reading taken from an instrument with a
systematic error will give an inaccurate result.

Random Errors
Try asking 10 people to read the level of liquid
in the same measuring cylinder. There will almost
certainly be small differences in their estimates
of the level. Connect a voltmeter into a circuit,
take a reading, disconnect the meter, reconnect
it and measure the same voltage again. There
might be a slight difference between the
readings.

These are random (unpredictable) errors. Random
errors can never be eliminated completely but we
can usually be sure that the correct reading lies
within certain limits.
To indicate this to the reader of the experiment
report, the results of measurements should be
written as

Result Uncertainty
16

For example, suppose we measure a length, l to
be 25 cm with an uncertainty of 0.1 cm. We write
the result as
By this, we mean that all we are sure about is
that is somewhere in the range 24.9 cm to 25.1
cm.

l 25.0 0.1cm
17
A. Quantifying the Uncertainty

The number we write as the uncertainty tells the
reader about the instrument used to make the
measurement. (As stated above, we assume that the
instrument has been used correctly.) Consider the
following examples.

Example 1 Using a ruler
The length of the object being measured is
obviously somewhere near 4.3 cm (but it is
certainly not exactly 4.35 cm). The result could
therefore be stated as
4.3 cm Half the smallest division on the ruler

In choosing an uncertainty equal to half the
smallest division on the ruler, we are accepting
a range of possible results equal to the size of
the smallest division on the ruler.
However, do you notice something which has not
been taken into account? A measurement of length
is, in fact, a measure of two positions and then
a subtraction.

Was the end of the object exactly opposite the
zero of the ruler? This becomes more obvious if
we consider the measurement again, as shown
below.

Notice that the left-hand end of the object is
not exactly opposite the 2 cm mark of the ruler.
It is nearer to 2 cm than to 2.05 cm, but this
measurement is subject to the same level of
uncertainty.
Therefore the length of the object is
(6.30 0.05)cm - (2.00 0.05)cm

so, the length can be between
(6.30 0.05) - (2.00 - 0.05) and (6.30 - 0.05)
- (2.00 0.05)
that is, between 4.40 cm and 4.20 cm
We now see that the range of possible results is
0.2 cm, so we write
Length 4.30 cm 0.10 cm
In general, we state a result as

Reading The smallest division on the measuring instrument
23

One may record the length of the following red
stick to be 5.9 0.1 cm.

Example 2 Using a Stop-Watch
Consider using a stop-watch which measures to
1/100 of a second to find the time for a pendulum
to oscillate once. Suppose that this time is
about 1s. Then, the smallest division on the
watch is only about 1 of the time being
measured. We could write the result as
T 1.00 0.01s
which is equivalent to saying that the time T is
between 0.99s and 1.01s.

This sounds quite good until you remember that
the reaction-time of the person using the watch
might be about 0.1s. Let us be pessimistic and
say that the person's reaction-time is 0.15s. Now
considering the measurement again, with a
possible 0.15s at the starting and stopping time
of the watch, we should now state the result as
T 1.00 s (0.01 0.3) s

In other words, T is between about 0.7s and
1.3s. We could probably have guessed the answer
to this degree of precision even without a
stop-watch!

Conclusions from the preceding discussion
If we accept that an uncertainty (sometimes
called an indeterminacy) of about 1 of the
measurement being made is reasonable, then

(a) a ruler, marked in mm, is useful for making measurements of distances of about 100mm (or 10 cm) or greater.
(b) a manually operated stop-watch is useful for measuring times of about 30 s or more (for precise measurements of shorter times, an electronically operated watch should be used)
28
B. How many Decimal Places?

Suppose you have a timer which measures to a
precision of 0.01s and it gives a reading of 4.58
s. The actual time being measured could have been
4.576 s or 4.585 s etc. However, we have no way
of knowing this, so we can only write
t 4.58s 0.01s

We now repeat the experiment using a better timer
which measures to a precision of 0.0001 s. The
timer might still give us a time of 4.58s but now
we would indicate the greater precision of the
instrument being used by stating the result as
t 4.5800 s 0.0001 s
So, as a general rule, look at the precision of
the instrument being used and state the result to
that number of decimal places.

30
C. How does an uncertainty in a measurement
affect the FINAL result?

The measurements we make during an experiment are
usually not the final result they are used to
calculate the final result. When considering how
an uncertainty in a measurement will affect the
final result, it will be helpful to express
uncertainties in a slightly different way.
Remember, the uncertainty in a given measurement
should be much smaller than the measurement
itself.

For example, if you write, "I measured the time
to a precision of 0.01s", it sounds good unless
you then inform your reader that the time
measured was 0.02s! The uncertainty is 50 of the
measured time so, in reality, the measurement is
useless.

We will define the quantity Relative Uncertainty
(sometimes called fractional uncertainty) as
follows
(To emphasize the difference, we use the term
"absolute uncertainty" where previously we simply
said "uncertainty").

Relative Uncertainty (Absolute Uncertainty) / (Measured Value)
33

Exercises
1. If we use the formula xy/z3 and the
percentage uncertainty (relative uncertainty
100) in y is 3 and in z is 4, what is it
percentage uncertainty in x?
2. Same as above, but the formula is xy2/vz ?
1). 15 2). 8

We will now see how to answer the question in the
title. It is always possible, in simple
situations, to find the effect on the final
result by straightforward calculations but the
following rules can help to reduce the number of
calculations needed in more complicated
situations.

35
Rule 1 If a measured quantity is multiplied or divided by a constant, then the relative uncertainty stays the same. See Example 1.
Rule 2 If two measured quantities are added or subtracted then their absolute uncertainties are added. See Example 2.
Rule 3 If two (or more) measured quantities are multiplied or divided then their relative uncertainties are added. See Example 3.
Rule 4 If a measured quantity is raised to a power then the relative uncertainty is multiplied by that power. (If you think about this rule, you will realise that it is just a special case of rule 3.) See Example 4.
36

A few simple examples might help to illustrate
the use of these rules. (Rule 2 has, in fact,
already been used in the section "Using a Ruler"
on page 3.)

Example 1
Suppose that you want to find the average
thickness of a page of a book. We might find that
100 pages of the book have a total thickness of T
9.0 mm. If this measurement is made using an
instrument having a precision of 0.1 mm, then the
relative uncertainty is e 0.1/9.0. Hence, the
average thickness of one page, t, is given by t
T/100 0.09 mm with an absolute uncertainty 0.09
x e 0.001mm, or t 0.090 mm 0.001mm. Note
both T and t have only 2 sf.

Example 2
(a) To find a change in temperature T T2-T1 ,
in which the initial temperature T1 is found to
be 20C 1C and the final temperature T2 is
found to be 45C 1C. Then T 25 2C.
(b) Now, the initial temperature T1 is found to
be 20.2C 0.1C and the final temperature T2 is
found to be 45.23C 0.01C. Then the calculated
value is 25.03 0.11. However, the least decimal
place measurement is 20.2 with only one decimal
point. So the answer is T 25.0 0.1C.

Exercise
3. The first part of the trip took 27 ? 3 (s),
the second part 14 ? 2 (s). How long time did the
whole trip take? How much longer did the first
part take compared to the second part?
41 ? 5s , 13 ? 5s

Example 3
To measure a surface area, S, we measure two
dimensions, say, x and y, and then use
S xy. Using a ruler marked in mm, we measure x
54 1 mm and y 83 1 mm. This means the
relative uncertainties of x and y are,
respectively, 1/54 and 1/83. The relative
uncertainty of S is then e 1/54 1/83 0.03.
The calculated value of the surface area is 4482
with uncertainty 134.46. Thus, the surface area
S is 4500 100mm². (2 significant figures)

Exercises
4. An object covers 433.07 ? 1.05 (m) in 23.09 ?
1.10 (s). What was the speed?
5. If using the formula v u at we insert u
6.0 ? 0.4 ms-1, a 0.200 ? 0.002 ms-2 and time t
2.00 ? 0.10 s, what will v be?

Example 4
To find the volume of a sphere, we first find its
radius, r (usually by measuring its diameter).
We then use the formula V 4/3 (p r3) . Suppose
that the diameter of a sphere is measured (using
an instrument having a precision of 0.2mm) and
found to be 50.0mm, so r 25.0mm with relative
uncertainty 0.2/50 0.004, so r 25.0 0.1mm.
The relative uncertainty of V is 3 x 0.004
0.012. The volume of the sphere is V 65500
800mm3.
(3 significant figures)

Exercises
6. The dimensions of piece of paper are measured
using a ruler marked in mm. The results were x
60mm and y 45mm.
(a) Rewrite the results of these measurements
"correctly".
(b) Calculate the maximum and minimum values of
the area of the sheet of paper which these
measurements give.
(c) Express the result of the calculation of
area of the sheet of paper in the form area A
A.
(a) 60 1mm, 45 1mm. (b) 2806mm2, 2596mm2.
(c) 2701 105mm2.

7. A body is observed to move a distance s 10m
in a time t 4 s. The distance was measured
using a ruler marked in cm and the time was
measured using a watch giving readings to 0.1 s
a) Express these results "correctly" (that is,
giving the right number of significant figures
and the appropriate indeterminacy).
b) Use the measurements to calculate the speed
of the body, including the uncertainty in the
value of the speed.
Distance 10 0.01 m, time 4 0.1s.
Since, v 2.5m/s. The uncertainty 0.065m/s.
Therefore, v 2.5 0.1 m/s.

8. A body which is initially at rest, starts to
move with acceleration a. It moves a distance s
12.00 0.12m in a time t 4.5 0.1s. Calculate
the acceleration.
1.2 0.1 m/s

9. The diameter of a cylindrical piece of metal
is measured to a precision of 0.02mm. The
diameter is measured at five different points
along the length of the cylinder. The results are
shown below. Units are mm.
(i) 7, (ii) 9.4, (iii) 5.6, (iv) 5 and (v)
4.8
(a) Rewrite the list of results "correctly".
(b) Calculate the average value of the diameter.
(c) State the average value of the diameter in a
way which gives an indication of the precision of
the manufacturing process used to make the
cylinder.
(d) Calculate the average value of the area of
cross section of the cylinder. State the result
as area A A(mm2).
(a) (i)7.00 0.02, (ii) 9.40 0.02, (iii) 5.60
0.02, (iv) 5.00 0.02, and (v) 4.80 0.02
(b) 6.36 mm.
(c) 0.02mm, 6.36 0.02mm.
(d) 31.77 0.20mm2.

47
3. Graphs

The results of an experiment are often used to
plot a graph. A graph can be used to verify the
relation between two variables and, at the same
time, give an immediate impression of the
precision of the results. When we plot a graph,
the independent variable is plotted on the
horizontal axis. (The independent variable is the
cause and the dependent variable is the effect.)

48
A. Straight Line Graphs

If one variable is directly proportional to
another variable, then a graph of these two
variables will be a straight line passing through
the origin of the axes. So, for example, Ohm's
Law has been verified if a graph of voltage
against current (for a metal conductor at
constant temperature) is a straight line passing
through (0,0). Similarly, when current flows
through a given resistor, the power dissipated is
directly proportional to the current squared. If
we wanted to verify this fact we could plot a
graph of power (vertical) against current squared
(horizontal). This graph should also be a
straight line passing through (0,0).

49
B. The "Best-Fit" Line

The best-fit line is the straight line which
passes as near to as many of the points as
possible. By drawing such a line, we are
attempting to minimize the effects of random
errors in the measurements. For example, if our
points look like this ?

The best-fit line should then be ?
Notice that the best-fit line does not
necessarily pass through any of the points
plotted.

51
C. To Measure the Slope of a Graph

The slope of a graph tells us how a change in one
variable affects the value of another variable.
The slope of the graph is defined as an must, of
course, be stated in the appropriate UNITS.

Slope vertical change / horizontal change
52

(x1, y1) and (x2, y2) can be the co-ordinates of
any two points on the line but for best
precision, they should be as far apart as
possible as shown in the two examples below.

In the second graph, it is clear that y decreases
as x increases so in this case, the slope is
negative.

54
D. Error Bars

Instead of plotting points on a graph we
sometimes plot lines representing the uncertainty
in the measurements. These lines are called error
bars and if we plot both vertical and horizontal
bars we have what might be called error
rectangles, as shown on next slide

x was measured to 0.5s, y was measured to 0.3m

The best-fit line could be any line which passes
through all of the rectangles. Assume that the
line passes through zero, use the example above
to estimate the maximum and minimum slopes of
lines which are consistent with these data. (The
diagram is too small to expect accurate answers
but you should find about 1.06ms-1 maximum and
about 0.92ms-1 minimum.)

57
E. Measuring the slope at a Point on a Curved
Graph

Usually we will plot results which we expect to
give us a straight line. If we plot a graph
which we expect to give us a smooth curve, we
might want to find the slope of the curve at a
given point for example, the slope of a
displacement against time graph tells us the
(instantaneous) velocity of the object.

To find the slope at a given point, draw a
tangent to the curve at that point and then find
the slope of the tangent in the usual way. As
shown, a tangent to the curve has been drawn at x
4.5s. The slope of the graph at this point is
given by ?y/?x (approximately) 5ms-1.

Exercises
1. The diameter of a metal ball is measured to be
28.0mm 0.2mm. The mass of the ball is measured
to be 120g 2g. Use these results to find a value
for the density r, of the metal of which the ball
is made.
Density is defined as mass per unit volume so to
calculate the density of a substance we use the
equation
0.0104 0.0004g/mm3

?m/V
60

2. An investigation was undertaken to determine
the relationship between the length of a pendulum
l and the time taken for the pendulum to
oscillate twenty times. The time it takes to
complete one swing back and forth is called the
period T. It can be shown that T 2p v(l /g)
where g is the acceleration due to gravity. The
following data was obtained.

61
Length of pendulum 0.05m Time for 20 oscillations 0.2s Period T T2 Absolute error of T2
0.20 17.5
0.42 24.8
0.59 32.0
0.81 35.9
1.02 41.0
62

(a) Complete the period column for the
measurements. Be sure to give the uncertainty and
the units of T.
(b) Calculate the various values for T2 including
its units.
(c) Determine the absolute error of T2 for each
value.
(d) Draw a graph of T2 against l. Make sure that
you choose an appropriate scale to use as much of
a piece of graph paper as possible. Label the
axes, put a heading on the graph, and use error
bars. Draw the curve of best fit.
(e) What is the relationship that exists between
T2 and l?
(f) Are there any outliers?
(g) From the graph determine a value of g.

3 Suppose the relationship between the semi-major
axis a and the sidereal period P of a planet in
the solar system is given by ,
where k and a are constants. From the table
below, plot a suitable graph to find k and a.

Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune
Semi-majorAxis a (AU) 0.3871 0.7233 1.0000 1.5237 5.2028 9.5388 19.1914 30.0611
SiderealPeriod P (year) 0.2408 0.6152 1.0000 1.8809 11.862 29.458 84.01 164.79
63
64
4. Writing Experimental Reports

You will perform a number of experiments in the
future. You must keep a record of ALL the
experiments which you perform. For a few of the
experiments you will be required to present a
full, detailed report, which will be graded. The
grades will form part of your final result
(remember, a significant percentage of your final
result will be based on your practical
abilities). Usually, a report is set out as
follows.

Title Introduction Diagram Method Results Theory Conclusion
65
1. Title (Aim)

The title must state clearly the aim of the
experiment. It must tell the reader what you are
trying to prove or measure. For example, Ohms
Law is not a suitable title for an experiment
report, whereas, Experiment to verify Ohms Law
is a suitable title. Similarly, Relative
Density is not a suitable title but Experiment
to measure the Relative Density of some common
substances is a more suitable title.

66
2. Introduction

If the experiment is designed to verify a law,
state the law in the introduction. The
introduction can also include such ideas as why
the results/conclusions of the experiment are
important in every-day life, in industry etc. (It
might even include a little historical
background, but not too much).

67
3. Hypothesis

Before starting your investigation you usually
have some idea of what you expect the results
will show. The hypothesis is basically a
statement of what you are expecting. If you are
trying, for example, to show how two variables
are related, state the expected relation and try
to give and explanation of you choice.

For example, when Newton was thinking about
gravitation, he assumed that the strength of the
force of gravity would become weaker as one moved
further from the body causing the field. He
suggested that the relation between force and
distance might involve the inverse of the
distance squared and to defend this choice he
pointed out that the surface area of a sphere
depends on the square of its radius. Various
observations then confirmed this suggestion.

69
4. Diagram

In most cases a labelled diagram is useful. Every
electrical experiment report must include a
circuit diagram. If diagrams are drawn by hand,
use a sharp pencil and a ruler. (If you use a
computer, learn how to make the best use of your
drawing program.)

70
5. Method

The method section should give enough detail to
enable another experimenter to repeat the
experiment to see if he/she agrees with your
results/conclusions. The method should include
- a description of the apparatus used
- what measurements you made (if possible, in
the order you made them)
- what precautions you took to ensure the best
accuracy possible
- a mention of any unexpected difficulties (and
how you overcame them)

71
6. Results

You should record all the measurements made
during the experiment along with some indication
of the uncertainty of each measurement. Whenever
possible, present the results in the form of a
table.

72
7. Theory

This section should include any information which
might help the reader to understand how you used
your measurements to reach the aim mentioned in
the title. For example, in one experiment,
designed to measure the relative density of a
substance, the actual measurements made are two
distances. The theory section of a report on such
an experiment must include a clear explanation of
how these two distances are related to the
relative density of the substance being
measured.

73
8. Conclusion/Evaluation

Every experiment report must have a conclusion.
If your aim was to verify a law, state whether
you have verified the law or not. If the aim was
to measure a particular quantity (e.g. relative
density), give the final measured value of the
quantity in the conclusion.
In the case of an experiment designed to measure
some well known physical constant you should
attempt to explain any difference between your
result and the accepted result.

For example, if you find g 9.5ms-2, you should
try to think of the most likely cause of this
obvious error.
If the experiment results were in some way
unsatisfactory, try to suggest how the
investigation might be improved in order to
improve accuracy of measurements or range of data
obtained. This evaluation section should include
comments on the apparatus used and the method
employed.

75
5. General Laboratory Rules

Do not enter the laboratory unless the laboratory
superintendent is present.
When you come into the laboratory, you should
walk to your place calmly. If you run you are
sure to bump into someone - and if he is carrying
equipment there could be an accident.
At your place, take out your writing materials,
file and text book. Put your bag under the bench
and put your coat out of the way in a clean area.
Do not leave anything lying beside your bench
someone is sure to fall over it.

Wear a white laboratory jacket. This is important
as the laboratory jacket will stop your clothes
from getting dirty or burnt.
If you have long hair make sure that you have an
elastic band or a hair clip to tie it back. This
will help to keep it out of the way which is much
safer.
Wear safety spectacles to protect your eyes when
you are using chemicals such as acids and when
you are boiling liquids or heating solids.

On your bench you find a water tap and a sink.
The laboratory is not as clean as a cafeteria, so
do not drink from the taps in the laboratory.
Also do not eat in the laboratory. After you have
been doing practical work, especially if you have
been handling animals or chemicals, wash your
hands carefully.
You will also find a gas tap on your bench. This
is for use with the Bunsen burner. Sometimes you
will need to heat things and that is what the
Bunsen burner is used for. Your teacher will show
you how to use one properly. When you are not
using a Bunsen burner the gas tap should be
turned off all the time.

The third thing that you will find on your bench
is an electricity socket. If you use the
electricity on your bench, for example when you
use a lamp, make sure that the bench is dry.
Some practical investigations are wet, so drops
of water can spread all over the bench and your
papers. It is a good idea to remove everything
that you do not need from your bench during
practical work. You will only need a pencil or a
pen to write with, a piece of paper for your
results and the instructions. The instructions
can be kept safe in a plastic folder.

When you do a practical investigation you will
need to collect equipment and materials. Never
carry too much equipment each time.
When you have finished a practical investigation
always leave your bench clean and dry. You can
rinse and clean your test tubes in the sink but
do not put solid objects down the sink, they will
block it.
When you leave the laboratory take all your
belongings with you and make sure that everything
is turned off (gas tap, water tap)
If you should have an accident always tell the
laboratory superintendent.

Exercises
Filling in the missing words.
1. Put all coats in a ...........................
.........
2. Your bags should be put under
.........................
3. Never .................................... or
shout in a laboratory.
4. Do not use the services (gas, water or
....................................) unless you
are told to.

5. Never .................................... or
drink in a laboratory
6. .................................. long hair
during practical work.
7. Protect your eyes with ......................
.............. when you boil liquids
8. Put all solid waste in the
................................. not in the
.................................

9. Put broken glass in the ......................
..............
10. Report all ..................................
.. immediately to the laboratory superintendent.
11. Wash your hands after handling ..............
........ and .................................
clean, out of the way area, the bench, run,
electricity, eat, tie back, safety spectacles,
plastic waste bin, sink, metal waste bin,
accidents, chemicals, animals.
End

83
Supplementary Notes

Indices and Logarithm
Scientists investigate the dependence of two or
more physical or chemical or biological
quantities. For example, the relationship between
a current I flowing through a light bulb with
resistance R and the potential difference V
across the light bulb is simply VIR. However,
physical situations are usually not simple,
mathematical tools, like indices and logarithms,
are required.

84
(a) am?an amn
(b) am/an am-n
(c) (am)n amn
(d) am ?bm (ab)m
(e) (a/b)m am/bm
(f) a0 1, where a ? 0
(g) a-m 1/ am, where a ? 0
(h) v(a2) a

Laws for Indices
If a, b are real numbers and m, n are positive
integers, we have the following laws

85
(a) loga 1 0
(b) loga a 1
(c) loga (MN) loga M loga N
(d) loga(M/N) loga M - loga N
(e) loga Mp p loga M
(f) loga N logb N/ logb a, where a ? 0 , N gt 0

Properties of Logarithm
Definition If ax N , then x loga N.

In physics, engineering and economics, the
natural logarithms are most often used. Natural
logarithms use the base e 2.71828, so that
given a number ex , its natural logarithm is x .
For example, e3.6888 is approximately equal to
40, so that the natural logarithm of 40 is about
3.6888. The usual notation for the natural
logarithm of x is ln(x) and for logarithms to the
base 10 is log(x).