PhaseIII Phy L1 ScienceSkill&Safety Ppt

8/12/2019 PhaseIII Phy L1 ScienceSkill&Safety Ppt

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Module: Science Skills andSafetyTime allocation: 8 hours

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IJSO Training Course

Phase III Objectives:

Describe, distinguish between randomuncertainties and systematic errors.Define and apply the concept of significantfigures.Identify and determine the uncertainties in resultscalculated from quantities and in a straight-linegraph.

Introduce general procedures of writingexperimental reports

Laboratory safety and rules.2

1. Significant Digits

Suppose you want to find the volume of alead cube. You could measure the length lof the side of a lead cube to be 1.76 cm andthe volume 13 from your calculator reads

5.451776. The measurement 1.76 cm was tothree significant figures so the answer canonly be three significant figures. So that thevolume = 5.45 cm3. The following rules areapplied generally:

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All non-zero digits are significant. (e.g., 22.2 has 3sf)

All zeros between two non-zero digits aresignificant. (e.g., 1007 has 4 sf)

For numbers less than one, zeros directly after the

decimal point are not significant. (e.g., 0.0024 has2 sf)A zero to the right of a decimal and following anon-zero digit is significant. (e.g., 0.0500 has 3 sf)

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

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For example, if you multiply 3.22 cm by 12.34 cmby 1.8 cm to find the volume of a piece of wood,you get an initial answer 71.52264 cm3 from yourcalculator. However, the least significantmeasurement is 1.8 cm with 2 sf. Therefore, thecorrect answer is only 72 cm3.

When adding and subtracting a series ofmeasurements, the answer has decimal places withthe least accurate place value in the series ofmeasurements.

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

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Exercises:

1. How many significant figures are indicated byeach 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). 38

2. (a) Add the following lengths of 3.15 mm and7.32 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 theanswer is 19.3 m.

(b) 2.34 x 90.66 x 3.7 = 784.93428. Thus theanswer is 780cm3.

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2. Making Measurements

A measurement should always be regarded as anestimate. The precision of the final result of anexperiment cannot be better than the precision ofthe measurements made during the experiment, sothe aim of the experimenter should be to make the

estimates as good as possible.

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There are many factors which contribute to theaccuracy of a measurement. Perhaps the mostobvious of these is the level of attention paid bythe person making the measurements: a carelessexperimenter gets bad results! However, if theexperiment is well designed, one carelessmeasurement will usually be obvious and cantherefore be ignored in the final analysis.

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

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

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Random Errors

Try asking 10 people to read the level of liquid inthe same measuring cylinder. There will almostcertainly be small differences in their estimates ofthe level. Connect a voltmeter into a circuit, take areading, disconnect the meter, reconnect it andmeasure the same voltage again. There might be aslight difference between the readings.

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These are random (unpredictable) errors. Randomerrors can never be eliminated completely but wecan usually be sure that the correct reading lieswithin certain limits.

To indicate this to the reader of the experiment

report, the results of measurements should bewritten as

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Result Uncertainty

For example, suppose we measure a length, l tobe 25 cm with an uncertainty of 0.1 cm. We writethe 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.

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l = 25.0 + 0.1cm

A. Quantifying the Uncertainty

The number we write as the uncertainty tells thereader about the instrument used to make themeasurement. (As stated above, we assume thatthe instrument has been used correctly.) Considerthe following examples.

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Example 1: Using a ruler

The length of the object being measured isobviously somewhere near 4.3 cm (but it iscertainly not exactly 4.35 cm). The result couldtherefore be stated as:

4.3 cm Half the smallest division on the ruler 18


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In choosing an uncertainty equal to half thesmallest division on the ruler, we are accepting arange of possible results equal to the size of thesmallest division on the ruler.

However, do you notice something which has notbeen taken into account? A measurement of lengthis, in fact, a measure of two positions and then asubtraction.

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

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Notice that the left-hand end of the object is notexactly opposite the 2 cm mark of the ruler. It isnearer to 2 cm than to 2.05 cm, but thismeasurement is subject to the same level ofuncertainty.

Therefore the length of the object is(6.300.05)cm - (2.000.05)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 cmWe now see that the range of possible results is0.2 cm, so we writeLength = 4.30 cm 0.10 cm

In general, we state a result as

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Reading The smallest division on the measuringinstrument

One may record the length of the following redstick to be 5.90.1 cm.

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Example 2: Using a Stop-WatchConsider using a stop-watch which measures to1/100 of a second to find the time for a pendulumto oscillate once. Suppose that this time is about1s. Then, the smallest division on the watch isonly about 1% of the time being measured. Wecould 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


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This sounds quite good until you remember thatthe reaction-time of the person using the watchmight be about 0.1s. Let us be pessimistic and saythat the person's reaction-time is 0.15s. Nowconsidering the measurement again, with a

possible 0.15s at the starting and stopping time ofthe watch, we should now state the result asT = 1.00 s (0.01+ 0.3) s

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

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Conclusions from the preceding discussion

If we accept that an uncertainty (sometimes calledan indeterminacy) of about 1% of themeasurement being made is reasonable, then

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(a) a ruler, marked in mm, is useful for making measurementsof distances of about 100mm (or 10 cm) or greater.

(b) a manually operated stop-watch is useful for measuringtimes of about 30 s or more (for precise measurements ofshorter times, an electronically operated watch should beused)

B. How many Decimal Places?

Suppose you have a timer which measures to aprecision of 0.01s and it gives a reading of 4.58 s.The actual time being measured could have been4.576 s or 4.585 s etc. However, we have no wayof knowing this, so we can only write

t = 4.58s0.01s

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We now repeat the experiment using a better timerwhich measures to a precision of 0.0001 s. Thetimer might still give us a time of 4.58s but nowwe would indicate the greater precision of theinstrument being used by stating the result as

t = 4.5800 s 0.0001 sSo, as a general rule, look at the precision of theinstrument being used and state the result to thatnumber of decimal places.

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C. How does an uncertainty in ameasurement affect the FINAL result?

The measurements we make during an experimentare usually not the final result they are used tocalculate the final result. When considering howan uncertainty in a measurement will affect thefinal result, it will be helpful to expressuncertainties in a slightly different way.Remember, the uncertainty in a givenmeasurement should be much smaller than themeasurement itself.

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For example, if you write, "I measured the time toa precision of 0.01s", it sounds good: unless youthen inform your reader that the time measuredwas 0.02s! The uncertainty is 50% of themeasured time so, in reality, the measurement isuseless.

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We will define the quantity Relative Uncertainty(sometimes called fractional uncertainty) asfollows

(To emphasize the difference, we use the term"absolute uncertainty" where previously wesimply said "uncertainty").

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

Exercises:

1. If we use the formula x=y/z3 and the percentageuncertainty (relative uncertainty 100%) in y is 3%and in z is 4%, what is it percentage uncertainty inx?

2. Same as above, but the formula is x=y2/z ?

1). 15% 2). 8%

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We will now see how to answer the question in thetitle. It is always possible, in simple situations, tofind the effect on the final result bystraightforward calculations but the followingrules can help to reduce the number of calculations

needed in more complicated situations.

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Rule 1: If a measured quantity is multiplied or divided by aconstant, then therelative uncertaintystays the same. SeeExample 1.

Rule 2: If two measured quantities areaddedor subtractedthentheirabsolute uncertaintiesareadded. SeeExample 2.

Rule 3: If two (or more) measured quantities aremultipliedordividedthen theirrelativeuncertainties areadded. SeeExample 3.

Rule 4: If a measured quantity israised to a powerthen therelative uncertainty ismultiplied by that power. (If youthink about this rule, you will realise that it is just a specialcase of rule 3.) SeeExample 4.

A few simple examples might help to illustrate theuse of these rules. (Rule 2 has, in fact, already

been used in the section "Using a Ruler" on page3.)

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Example 1

Suppose that you want to find the averagethickness of a page of a book. We might find that100 pages of the book have a total thickness of T= 9.0 mm. If this measurement is made using aninstrument having a precision of 0.1 mm, then therelative uncertainty is e= 0.1/9.0. Hence, theaverage thickness of one page, t, is given by t =T/100 = 0.09 mm with an absolute uncertainty0.09 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 be20C1C and the final temperature T2 is foundto be 45C1C. Then T = 25 2C.

(b) Now, the initial temperature T1 is found to be20.2C 0.1C and the final temperature T2 isfound to be 45.23C0.01C. Then the calculatedvalue is 25.03 0.11. However, the least decimal

place measurement is 20.2 with only one decimalpoint. So the answer is T = 25.0 0.1C.

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Exercise:3. The first part of the trip took 273 (s), thesecond part 142 (s). How long time did thewhole trip take? How much longer did the first

part take compared to the second part?

41 5s , 135s

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Example 3To measure a surface area, S, we measure twodimensions, say, x and y, and then useS = xy. Using a ruler marked in mm, we measurex = 54 1 mm and y = 831 mm. This means the

relative uncertainties of x and y are, respectively,1/54 and 1/83. The relative uncertainty of S isthen e = 1/54 + 1/83 = 0.03. The calculated valueof the surface area is 4482 with uncertainty134.46. Thus, the surface area S is 4500100mm. (2 significant figures)

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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 = 6.00.4 ms-1, a = 0.200 0.002 ms-2 and time t =2.00 0.10 s, what will v be?

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Example 4

To find the volume of a sphere, we first find itsradius, r (usually by measuring its diameter). Wethen use the formula: V = 4/3 ( r3) . Suppose thatthe diameter of a sphere is measured (using aninstrument having a precision of 0.2mm) andfound to be 50.0mm, so r = 25.0mm with relativeuncertainty 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.

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Exercises:

6. The dimensions of piece of paper are measuredusing 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 ofthe area of the sheet of paper which thesemeasurements give.(c) Express the result of the calculation of area of thesheet of paper in the form: area = A A.

(a) 60 1mm, 451mm. (b) 2806mm2, 2596mm2.(c) 2701 105mm2.

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7. A body is observed to move a distance s = 10min a time t = 4 s. The distance was measured usinga ruler marked in cm and the time was measuredusing a watch giving readings to 0.1 s

a) Express these results "correctly" (that is, givingthe right number of significant figures and theappropriate indeterminacy).

b) Use the measurements to calculate the speed ofthe body, including the uncertainty in the value ofthe 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 movewith acceleration a. It moves a distance s = 12.000.12m in a time t = 4.5 0.1s. Calculate theacceleration.

1.20.1 m/s

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9. The diameter of a cylindrical piece of metal is measured to aprecision of 0.02mm. The diameter is measured at five differentpoints along the length of the cylinder. The results are shown below.Units are mm.

(i) 7, (ii) 9.4, (ii i) 5.6, (iv) 5 and (v) 4.8

(a) Rewrite the li st of results "correctly".

(b) Calculate the average value of the diameter.

(c) State the average value of the diameter in a way which gives anindication of the precision of the manufacturing process used to makethe 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 0.20mm2.46

3. Graphs

The results of an experiment are often used to plota graph. A graph can be used to verify the relation

between two variables and, at the same time, givean immediate impression of the precision of theresults. When we plot a graph, the independentvariable is plotted on the horizontal axis. (Theindependent variable is the cause and thedependent variable is the effect.)

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A. Straight Line GraphsIf 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 againstcurrent (for a metal conductor at constant temperature) is astraight line passing through (0,0). Similarly, when currentflows through a given resistor, the power dissipated isdirectly proportional to the current squared. If we wantedto verify this fact we could plot a graph of power (vertical)against current squared (horizontal). This graph shouldalso be a straight line passing through (0,0).

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B. The "Best-Fit" LineThe best-fit line is the straight line which passes

as near to as many of the points as possible. Bydrawing such a line, we are attempting tominimize the effects of random errors in themeasurements. For example, if our points looklike this

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The best-fit line should then be

Notice that the best-fit line does not necessarilypass through any of the points plotted.

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C. To Measure the Slope of aGraph

The slope of a graph tells us how a change in onevariable affects the value of another variable. Theslope of the graph is defined as an must, of course,

be stated in the appropriate UNITS.

Slope = vertical change / horizontal change

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

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

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D. Error Bars

Instead of plotting points on a graph wesometimes plot lines representing the uncertaintyin the measurements. These lines are called error

bars and if we plot both vertical and horizontalbars we have what might be called errorrectangles, as shown on next slide:

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x was measured to 0.5s, y was measured to

0.3m

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The best-fit line could be any line which passesthrough all of the rectangles. Assume that the line

passes through zero, use the example above toestimate the maximum and minimum slopes oflines which are consistent with these data. (Thediagram is too small to expect accurate answers

but you should find about 1.06ms-1 maximum andabout 0.92ms-1 minimum.)

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E. Measuring the slope at a Pointon a Curved Graph

Usually we will plot results which we expect togive us a straight line. If we plot a graph whichwe expect to give us a smooth curve, we mightwant 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.

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

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Exercises:1. The diameter of a metal ball is measured to be28.0mm 0.2mm. The mass of the ball ismeasured to be 120g 2g. Use these results tofind a value for the density r, of the metal ofwhich the ball is made.Density is defined as mass per unit volume so tocalculate the density of a substance we use theequation:

0.01040.0004g/mm3

=m/V

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2. An investigation was undertaken to determinethe relationship between the length of a

pendulum l and the time taken for the pendulumto oscillate twenty times. The time it takes tocomplete one swing back and forth is called the

period T. It can be shown that T = 2 (l /g)where g is the acceleration due to gravity. Thefollowing data was obtained.

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Length of

pendulum0.05m

Time for 20

oscillations0.2s

PeriodT

T2 Absolute

error ofT2

0.20 17.5

0.42 24.8

0.59 32.0

0.81 35.9

1.02 41.0

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(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 youchoose an appropriate scale to use as much of a piece ofgraph paper as possible. Label the axes, put a heading onthe 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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Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

Semi-major

Axisa(AU)

0.3871 0.7233 1.0000 1.5237

5.2028 9.5388 19.1914 30.0611

SiderealPeriodP

(year)

0.2408 0.6152 1.0000 1.8809

11.862 29.458 84.01 164.79

3Suppose the relationship between the semi-majoraxisaand the sidereal periodPof a planet in thesolar system is given by , wherekand aare constants. From the table below, plot a suitablegraph to findkand a.

a kP=

4. Writing ExperimentalReports

You will perform a number of experiments in thefuture. You must keep a record of ALL theexperiments which you perform. For a few of theexperiments you will be required to present a full,detailed report, which will be graded. The gradeswill form part of your final result (remember, a

significant percentage of your final result will bebased on your practical abilities). Usually, areport 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 theexperiment. It must tell the reader what you aretrying to prove or measure. For example, OhmsLaw is not a suitable title for an experimentreport, whereas, Experiment to verify OhmsLaw is a suitable title. Similarly, RelativeDensity is not a suitable title but Experiment tomeasure the Relative Density of some commonsubstances is a more suitable title.

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2. Introduction:

If the experiment is designed to verify a law, statethe law in the introduction. The introduction canalso include such ideas as why theresults/conclusions of the experiment areimportant in every-day life, in industry etc. (Itmight even include a little historical background,

but not too much).

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When you do a practical investigation you willneed to collect equipment and materials. Nevercarry too much equipment each time.

When you have finished a practical investigationalways leave your bench clean and dry. You canrinse and clean your test tubes in the sink but donot put solid objects down the sink, they will

block it.When you leave the laboratory take all your

belongings with you and make sure thateverything is turned off (gas tap, water tap)

If you should have an accident always tell thelaboratory 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 alaboratory.

4. Do not use the services (gas, water or............................... .....) unless you are told to.

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5. Never ............................ ........ or drink in alaboratory

6. ................................ .. long hair during practicalwork.

7. Protect your eyes with ....................................

when you boil liquids8. Put all solid waste in the ................................ .not in the .................................

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9. Put broken glass in the ....................................

10. Report all ............................. ....... immediatelyto 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, plasticwaste bin, sink, metal waste bin, accidents,chemicals, animals.

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Supplementary Notes

Indices and LogarithmScientists investigate the dependence of two ormore physical or chemical or biological quantities.For example, the relationship between a current Iflowing through a light bulb with resistance R andthe potential difference V across the light bulb issimply V=IR. However, physical situations areusually not simple, mathematical tools, likeindices and logarithms, are required.

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Laws for IndicesIf a, b are realnumbers and m, n are

positive integers, wehave the followinglaws:

(a) am an = am+n

(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) (a2) = |a|

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Properties ofLogarithmDefinition: If ax = N ,then x = loga N.

(a) loga 1 = 0

(b) loga a = 1

(c) loga (MN) = logaM+ logaN

(d) loga(M/N) = logaM- logaN

(e) logaMp= plogaM

(f) logaN = logbN/ logb a, where a 0 , N > 0

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In physics, engineering and economics, the naturallogarithms are most often used. Naturallogarithms use the base e = 2.71828, so thatgiven a number ex , its natural logarithm is x . Forexample, e3.6888 is approximately equal to 40, sothat the natural logarithm of 40 is about 3.6888.The usual notation for the natural logarithm of x isln(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) + ln(4)

x = 5/7

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PhaseIII Phy L1 ScienceSkill&Safety Ppt

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