section i 2 measurement techniques

7/28/2019 Section I 2 Measurement Techniques

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Measurement Techniques

Content

2.1 Measurements 2.2 Errors and uncertainties

Learning Outcomes

(a) use techniques for the measurement of length, volume, angle,mass, time, temperature and electrical quantities appropriate to theranges of magnitude implied by the relevant parts of the syllabus. In

particular (1) measure lengths using a ruler, vernier scale and micrometer.

(2) measure weight and hence mass using spring and leverbalances.

(3) measure an angle using a protractor.

(4) measure time intervals using clocks, stopwatches and thecalibrated time-base of a cathode-ray oscilloscope (c.r.o).

(5) measure temperature using a thermometer as a sensor.

(6) use ammeters and voltmeters with appropriate scales.

(7) use a galvanometer in null methods.

(8) use a cathode-ray oscilloscope (c.r.o).

(9) use a calibrated Hall probe.


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* (b) use both analogue scales and digital displays.

* (c) use calibration curves.

(d) show an understanding of the distinction between

systematic errors (including zero errors) and random

errors. (e) show an understanding of the distinction between

precision and accuracy.

* (f) assess the uncertainty in a derived quantity by simple

addition of actual, fractional or percentage uncertainties (a

rigorous statistical treatment is not required)


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Introductionmeasurement & errors

All experiments are designed to obtain a quantitativeresult for a physicalquantity and it involve measurements

These measurements must be some combination of the base quantities

Measurements will be limited by the instruments available in the laboratory

Techniques must be acquired for the measurements of : length, volume,angle, mass, time, temperatureand electrical quantities.

It has to be assumed that the instruments are calibrated through

comparison, but the more difficult issue is determining which is the correct

instrument!

In the course of taking such measurements, errors are inevitable. Error in a scientific measurement means, the inevitable uncertainty that

occurs, in all measurements arising from the limitations of the observer, the

measuring instrument used or the methods used or a combination


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Instruments & methods, of some measurements Length

the metre rule the micrometer screw gauge

the vernier caliper

Pressure

manometer

barometer Mass

the top pan balance

the spring balance

the lever balance

current balance Angle

protractor


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Time

the stop-clock

digital timer

Frequency

cathode ray oscilloscope, frequency counter

Temperature mercury-in-glass thermometer

the thermocouple thermometer

Current & Voltage

analogue meters

galvanometer digital meter, multi-meter

cathode ray oscilloscope

Magnetic flux density

Hall probe


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Methods of measuring length

The metre rule

Simplest length-measuring instrument is the metre or half metre rule (i.e

100 cm or 50 cm)

Smallest division on the metre rule is 1 mm

Should be able to take a reading with an uncertainty of 0.5 mm

It is good practice to always check the wooden or plastic ruler with anengineers steel rule if one is available

Should be aware of 3 possible errors

End of the rule is worn out, giving an end error leading to something

called a systematic error

Calibration of the metre rule i.e. markings on the ruler are not accurate

Parallax error


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The vernier caliper

Very versatile instrument used for measuring the dimensions of an

object, the dimensions of a hole, or the depth of a hole

Range is up to 100 mm (10 cm) and it can be read to 0.1 mm (0.01 cm)

or 0.05 mm depending on type of vernier Should be aware of

Systematic zero error - very likely


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The micrometer screw gauge Used to measure the dimensions of objects up to a maximum of about 50 mm (5

cm)

On the barrel there are divisions every 0.5 mm

The screw advances exactly 1 mm for 2 revolutions i.e. the pitch of the screw is 0.5

mm or 500 m. Each revolution is 50 intervals

The graduations around the thimble run from 0 to 50 i.e. each division corresponds

to one hundredth of a mm, 0.01 mm or 10 m (0.001 cm)

The reading on the thimble is added to the reading on the barrel

If the micrometer is in doubt, it can be checked against a series of gauge blocks

Should be aware of

Systematic zero error - very likely


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Measurement of pressure

Difference in air/gas pressure can be measured by comparing the heights of

a liquid in the 2 arms of a U-tube. Instrument is called a manometer

The pressure of the gas is the difference in height, since pressure = hg

Aneroid barometer can be used to directly read pressure

Should be careful to read the meniscus

if concave, read the bottom of the meniscus

if mercury, read the convex or top part of the meniscus

to ensure tube is vertical

of parallax error


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Measurement of mass Instruments used are top-pan balance, lever balance and the spring balance

A balance compares the weight of the unknown mass with the weight of thestandard mass since weight is proportional to mass

Spring balance measures directly in force units i.e. Newton, but sometimesin kilograms

Top-pan balance ensure that the initial (unloaded) reading is zero

there is a control for adjusting the zero reading, balance may have a tare facilityi.e. mass of material added to the container is obtained directly

uncertainty will be quoted by manufacturer in the manual, usually as apercentage of the reading shown on the scale

The spring balance based on Hookes Law which states that extension is proportional to the load;

measurement is made directly by a moving over a circular scale

should be careful of zero error, usually has a zero error adjustment screw

parallax error

Lever balances based on principle of moments where unknown mass is balanced by a slider,

calibrated in mass units

should be aware of zero error, parallax error


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Measurement of angle

Protractor


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Measurement of time

The stop-clock

Digital timer


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Measurement of frequency

Cathode ray oscilloscope

Frequency counter


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Measurement of temperature

Mercury-in-glass thermometer

Thermocouple


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Measurement of current & voltage

analogue meters

galvanometer

digital meters

multi-meters

cathode ray oscilloscope


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Measurement of magnetic flux density

The Hall effectwhen current flows through a semiconductor in a

magnetic field, the charge carriers(electrons) will experience a force and a

potential difference known as the Hall voltage will develop across the

semiconductor

The Hall probe


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Errors, terminology & types

Basically errors can be divided into two major types:

(a) round-off errors

Round-off errors are due to the rounding off of non-terminating

numbers, like , to a certain number ofdecimal places or

significant figures. (b) measurement errors.

Measurement errors are further divided into systematic and

random errors.


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Absolute, Relative and Percentage Errors

Ifa is the actual/mean value of a quantity and x is a measured value, thenthe absolute error, ex, is = xa.

The relative error, rx is = ex/a.

The percentage relative error %rx= rx x 100%.

For example, if an object has an actual length of 47 cm, while its measuredlength is 48 cm, then :

ex= |4847| = 1 cm

rx= 1/47 = 0.021 3 sig fig

%rx= 0.02127 x 100 = 2% 1 sig fig


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Limits of Errors

The absolute error bounds are the limits between which the absolute errorlies. So, ifexwhere 0, then is called the maximum absolute errorbound ofx.

i.e. a =xwhere (x - )and (x+ )are called the lower and upper bounds fora.

In general, the maximum absolute error boundfor a number measuredor rounded-off to a degree of accuracy ofn decimal places is ( x10-n).

examples:

(1) A 50 kg cement bag measured to the nearest kg has a maximumabsolute error bound of 0.5 kg ( x 10-0) which gives it a measuredmass of (50.0 0.5) kg.

(2) Ifis rounded-off to 3.14, then its value is (3.140 0.005) [ = x10

-2

]. The limit of precision of a measuring device is thus the smallestdivision of measurement the device is able to display.


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Worked Example

An object weighs 3.2 kg on a weighing scale which is able to measure to

the nearest 0.1 kg.

Find the maximum value of the absolute error, the relative error and the

percentage error?

Solutiondegree of accuracy = x 10-1

Absolute error = x 0.1 = 0.05 kgMaximum relative error = 0.05/3.2 = 0.016

Maximum % error = 0.016 x 100% = 1.6 %


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Measurement Errors

Measurement errors arise because of the limit in the accuracy of the

instrument used and the physical constraints of the observer.

Measurement errors can be classified into two types:

systematic and random errors.

A systematic error causes a measurement to be consistently shifted in onedirection only (i.e. always too high or too low from its actual).

A random error has an equal chance of being too high or too low


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Systematic Errors

Systematic errors in the measurement of physical quantities areuncertainties that are due to:

(a) instruments faults;

(b) faults in the surrounding conditions or

(c) mistakes made by the observer.

Systematic errors are not revealed or eliminated by repeated

measurements Can be eliminated by calibration of the instrument used and by using

correct experimental and measurement techniques.


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Sources of systematic errors

The sources of systematic errors may be classified into thefollowing:

(a) Zero Errors

Zero errors occur if the reading on an instrument is not zero evenwhen it is not used to make any measurements. For examples:(i) a stopwatch does not point to zero when it has been reset; (ii)the micrometer screw gauge does not read zero when its gap isclosed.

(b) Personal Error of Observer

This error results from the physical constraints or limitation of anindividual: for example, the reaction time to an event which hasstarted a fraction of a second earlier.

(c) Errors due to Instruments

For examples: (i) A watch which is fast; (ii) An ammeter which isused under different conditions from which it had beencalibrated.

(d) Errors due to Wrong Assumptions

For example, the value ofgis assumed wrongly


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Random Errors

Random errors are uncertainties in a measurement made by the

observer or person who takes the measurement.

The characteristic of a random error is that it can be positive or

negative in absolute sense and its magnitude is not constant.

Random errors are unavoidable.

They can be minimised by taking a large number of measurementsunder the same conditions and averaging the readings.

Sources of Random Errors

Errors due to parallax when reading a scale at an angled

position.

Errors due to different pressures being applied when using amicrometer screw gauge to measure the diameter of a wire.

Errors due to changes in temperature while taking

measurements.


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Compound Errors

When evaluating or determining a particular physical quantity through

measurements of various other physical quantities related to it, errors for

the quantity are compounded and enlarged from errors made in the latter

quantities.


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Addition or Subtraction

Suppose that a physical quantity Uis related to two otherphysical quantitiesx andy as follows:

U=x +y

Ifx and y are the uncertainties in the measurement ofx andy

respectively, then the uncertainty in the determination ofUis:U= (x + y)

Also, ifV=xy, then the error in the determination ofVis:

V= ( x + y)


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Worked Example

The internal diameterd1 and the external diameterd2 of a metal tube are

d1 = (45 1) mm and d2 = (60 2) mm.

What is the percentage uncertainty in the thickness of the tube ?

Solution

Thickness of the tube, t= d2d1=(6045) mm = 15 mmUncertainty in t, t= (d1+d2) = (2+1) mm = 3 mm

percentage uncertainty in t:

t/tx 100% = 3/15 x 100% = 20%


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Product

Suppose that U=xyzwith errors inx, yand zas x, yand zrespectively.

Then the maximum fractional error in Uis:

U/U= [ x/x+ y/y+ z/z]

Worked Example

The dimensions of a box are recorded as follows:

length, l = (5.0 0.2) cmwidth, b = (4.0 0.1) cmheight, h = (8.0 0.2) cm

What is the maximum percentage error for the volume of the box ?

Solution

Volume of box = lx b x h = (5.0 x 4.0 x 8.0) cm = 160 cm

maximum percentage error in V:V/Vx 100% = [ l/l+ b/b + h/h ] x 100%

= [ 0.2/5.0 + 0.1/4.0 + 0.2/8.0 ] x 100%= 9 %


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Quotient

Suppose that U=x/y. Ifx and y are uncertainties inx andyrespectively

Then the maximum fractional error in Uis:

U/U= [x/x + y/y]

Worked Example

The mass of a block of metal is (11.5 0.5) kg and its volume is (1000

20) cm. How would you express the density of the metal?

Solution

Density, = M/V = 11.5/1000 = 1.15 x 10-2 kg/cm

Maximum fractional error in :

/=

[M/M+

V/V] =

[0.5/11.5 + 20/1000] = [0.5/11.5 + 20/1000] x (1.15 x 10-2)

= (0.07 x 10-2) kg/cm density of metal, = (1.15 0.07) x 10-2 kg/cm


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Example The density of a metal cube is given by = mass/volume and is

calculated from the following data:mass = 12.2 0.1 gcube side l= 2.1 0.05 cmWhat is the maximum possible error in (a) as a percentage ofand (b)absolute?

Solution

% V= 3(%l) =3 x ( 0.05/2.1 x 100%) = 7.1%%= (%M+ %V) = (0.1/12.2 x 100 + 7.1)%

= (0.82 + 7.1)%= 7.92% = 8%

= mass/volume = 12.2/2.13

= 1.3 g/cm

maximum absolute error = 8/100 x = 8/100 x (12.2/2.13)

= 0.106 g/cm

= 1.3

0.1 g/cm


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Example

Measurementsxand ywere given correct to two decimal places, as

x= 0.72 and y= 0.13.

Find the maximum value ofx 2y?

Solution

x

2y=x

y

yMaximum error in each x and y = 0.5 x 10-2 = 0.005

(x-2y) = 0.005 + (2 x 0.005) = 0.015

Maximum value of x 2y = 0.72 (2 x 0.13) + 0.015

= 0.475


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Example

A rectangular field is measured as 212 m by 189 m, both distances being

given to the nearest metre. What is the maximum error in the area of the

field ?

Solution

Relative errors of both dimensions (ex/a ) are 0.5/212 and 0.5/189 due tomeasurements to the nearest metre

(ex = x 100 )

maximum relative error in area = 0.5/212 + 0.5/189 = 0.005

maximum error = 0.005 x 212 x 189 = 200.34 = 200 m


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Precision, tolerance and accuracy

A precise measurement is one in which random errors are small.

Precision refers to the extent or limit of sensitivity of a given measuring

instrument to obtain the readings of the physical quantity being measured.

It also indicates the agreement among several measurements that have been

made in the same way. example, readings of the same quantity that are close to each other are

considered precise.


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Precision, tolerance and accuracy

Tolerance is a measure of the degree of precision of the device. In the caseof a stopwatch, an uncertainty of 0.1 s is commonly used, while for themetre rule 0.1 cm (1 mm), vernier caliper scale 0.01 cm (0.1 mm) andmicrometer screw gauge 0.001 cm (0.01 mm).

An accurate measurement is one in which systematic errors are small.

Accuracy refers to the degree of closeness of a measured quantity to itstrue value. It is therefore given as a percentage error.

Accuracy depends on the magnitude of the measured quantity and theprecision of the instrument used.


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Accurate experiment

An accurateexperiment is one for which the majority of the measuredvalues of a quantity are close to the true value.

A preciseexperiment is one which has a narrow spread of measured valuesbetween the lower and upper limit of values.

As such, precise readings are not necessarily accurate readings. example, if the zero reading of a micrometer screw gauge is not noted and

taken care of, measurements made using the gauge may be precise but

inaccurate. The readings may be higher (for positive zero error) or lower

(for negative zero error).


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Significant Digits/Figures

Significant figures are those digits in a measurement that are known withcertainty plus the first digit that is uncertain.

The degree of accuracy of a measured quantity can only be to the greatestdegree an instrument is able to measure. As such, any digit beyond thatcan be measured is irrelevant or insignificant.

When stating any measured quantity, the following rules should apply:

Experimental uncertainties should almost always be rounded to onesignificant figure. For example, g = 9.82 0.03 m/s and not 9.82 0.03385 m/s.

The last significant figure should usually be of the same order ofmagnitude as the uncertainty. For example x = 6050 30 m/s and not6051.78 30 m/s.

If the last significant digit is a 1 or 2 in a result, stating the answer withone extra figure trailing the digit is more appropriate. For example, thestated answer to calculated value 1415 correct to one significant figureshould be 1400 and not 1000.


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Significant Digits in Addition and Subtraction

When adding or subtracting measured quantities, the precision of the

answer can only be as great as the least precise termin the sum ordifference. All digits up to this limit of precision are significant.

Worked Example

Ifx = 3.76, y = 46.855 and z = 0.2. What is the correct value ofx + y + z?Solution

x + y + z = 3.76 + 46.855 + 0.2 = 50.815

largest value = 3.765 + 46.8555 + 0.25 = 50.8705

smallest value = 3.755 + 46.8545 + 0.15 = 50.7595

limit of precision 0.05 from x 10-1

= 50.870550.7595 = 0.1110/2 = 0.0555 0.05Since is greater than the limit of precision (0.05) for a one decimal

placing, the third digit ( .8 ) of the answer is itself uncertain.

correct value ofx + y + z = 3.76 + 46.855 + 0.2 = 50.815

50.8


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Significant Digits in Multiplying and Dividing

When multiplying or dividing measured quantities, the number of

significant digits in the result generally can be only as great as the least

number of significant digits in any factor in the calculation.

Example

Ifl = 31.3, b = 28 and h = 51.85, what is the value ofl x b x h?Solution

l x b x h = 31.3 x 28 x 51.85 = 45,441.34 45,000


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Exercises Measurements x, y and z are given correct to one decimal place as x = 2.12, y =

30.4, z = 9.7. Find the maximum errors in the following

(a) 2x + 3y 4z (b) xy (c) (x + y)/z

The four stages of a relay race take 21.3, 20.7, 19.5 and 23.1, all measurementsbeing given in seconds correct to one decimal place. What was the least possibletotal time taken?

A distance of 53 miles (to the nearest mile) was driven at 55 m.p.h. ( to the nearest 5m.p.h. ). What was the greatest and least values of the time taken.

The sides of a cuboid are 20.1, 19.3, 25,6, each measurement being given in

centimetres correct to one decimal place. Find the maximum values of the volumeand the surface area.

The density of the material of a rectangular block was determined by measuring themass and linear dimensions of the block. The the results obtained, together with theiruncertainties are

mass = ( 25.0 0.1 ) g ; length = ( 5.00 0.01 ) cmbreadth = ( 2.00 0.01 ) cm ; height = ( 1.00 0.01 ) cm

Calculate the density and uncertainty in this result. [ 2.50 0.05 ] A student makes measurements from which he calculates the speed of sound as

327.66 m/s. He estimates that his result is accurate only to 3%. What should be thespeed to be reported expressed to the appropriate number of significant figures?

[ 330 m/s ]


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Choice of instrument, method etc

Consider the nature of the measurement you have to make

Consider the accuracy

Consider the sensitivity

With direct reading measurements , readings can be obtained quickly and

accurately Availability of the instruments

Economics etc

In general all instruments should be kept clean and dry

Always use a container for lose objects or chemicals

section i 2 measurement techniques

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