Transcript Basic Relativity

IJSO Training Course
Phase III
Module: Science Skills and
Safety
Time allocation: 8 hours
1
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.
2
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:
3
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)
4
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.
5
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.
6
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.
7
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
8
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.
9
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.
10
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.
11
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.
12
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.
13
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.
14
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
15
For example, suppose we measure a length, l to
be 25 cm with an uncertainty of 0.1 cm. We write
the result as l = 25.0 + 0.1cm
By this, we mean that all we are sure about is that
is somewhere in the range 24.9 cm to 25.1 cm.
16
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.
17
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 18
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.
19
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.
20
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
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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
22
instrument
One may record the length of the following red
stick to be 5.9 ± 0.1 cm.
23
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.
24
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
25
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 stopwatch!
26
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)
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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
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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.
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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.
30
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.
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We will define the quantity Relative Uncertainty
(sometimes called fractional uncertainty) as
follows
Relative Uncertainty =
(Absolute Uncertainty) / (Measured Value)
(To emphasize the difference, we use the term
"absolute uncertainty" where previously we
simply said "uncertainty").
32
Exercises:
1. If we use the formula x=y/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 x=y2/√z ?
1). 15% 2). 8%
33
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.
34
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.
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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.)
36
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.
37
Example 2
(a) To find a change in temperature T = T2-T1 ,
in which the initial temperature T1 is found to be
20°C ± 1°C and the final temperature T2 is found
to be 45°C ± 1°C. Then T = 25 ± 2°C.
(b) Now, the initial temperature T1 is found to be
20.2°C ± 0.1°C and the final temperature T2 is
found to be 45.23°C ± 0.01°C. 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 ± 0.1°C.
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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
39
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)
40
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?
41
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 (π 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)
42
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.
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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.
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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
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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.
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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.)
47
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).
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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 
49
The best-fit line should then be 
Notice that the best-fit line does not necessarily
pass through any of the points plotted.
50
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
51
(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.
52
In the second graph, it is clear that y decreases as
x increases so in this case, the slope is negative.
53
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:
54
x was measured to ±0.5s, y was measured to ±
0.3m
55
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.)
56
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.
57
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 Δy/Δx =
(approximately) 5ms-1.
58
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: ρ=m/V
0.0104 ±0.0004g/mm3
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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 = 2π √(l /g)
where g is the acceleration due to gravity. The
following data was obtained.
60
Length of
pendulum
±0.05m
0.20
Time for 20
oscillations
±0.2s
17.5
0.42
24.8
0.59
32.0
0.81
35.9
1.02
41.0
Period T2 Absolute
T
error of T2
61
(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.
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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 a  kP , 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
Semimajor
Axis a
(AU)
0.3871
0.7233
1.0000
1.523
7
5.2028
9.5388
19.1914
30.0611
Sidereal
Period P
(year)
0.2408
0.6152
1.0000
1.880
9
11.862
29.458
84.01
164.79
63
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
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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, “Ohm’s
Law” is not a suitable title for an experiment
report, whereas, “Experiment to verify Ohm’s
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.
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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).
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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.
67
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.
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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.)
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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
70
how you overcame them)
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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 .................................
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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.
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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 V=IR. However, physical situations are
usually not simple, mathematical tools, like
indices and logarithms, are required.
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(a) am٠an = am+n
(b) am/an = am-n
Laws for Indices
If a, b are real
numbers and m, n are
positive integers, we
have the following
laws:
(c) (am)n = amn
(d) am ٠bm = (ab)m
(e) (a/b)m= am/bm
(f) a0 = 1, where a ≠ 0
(g) a-m =1/ am, where a ≠ 0
(h) √(a2) = |a|
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Properties of
Logarithm
Definition: If ax = N ,
then x = loga N.
(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>0
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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).
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Example 1.1:
It is known that Y =kXn . From the graph given,
find k and n.
n = -3/7 and k = 1000
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Example 1.2:
Solve
ln(x2 – 3x + 2) = 2 ln(2x - 1) + ½ ln(4)
x = 5/7
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