Measurement and Errors 2020

&

EP015

PHYSICS SEMESTER 1

KOLEJ MATRIKULASI TEKNIKAL JOHOR
MINISTRY OF EDUCATION, MALAYSIA
82000 PONTIAN, JOHOR
www.kmtj.matrik.edu.my

MEASUREMENT & ERRORS

1.0 Introduction :

Measurements are trials to determine the actual value of a particular physical quantity. The difference between
the actual value of a quantity and the value obtained in measurement is called error.

To measure is to make an acceptable estimate. A suitable and even the best measuring instrument (apparatus)
must be determined to be used.

2.0 Properties of Measurement :

A measurement depends on following properties :

1. Accuracy of the apparatus.
= An ability of the apparatus to give readings close (or almost equal) to the actual value of a
quantity.

2. Sensitivity of the apparatus.
= An ability of the apparatus to respond (or to detect) a small change in the value of a
measurement.

3. Consistency of the apparatus.
= An ability of the apparatus to register the same (almost the same) reading when a measurement
is repeated.

High consistency → Small deviation from the mean value.

Example :

Equipment Range Sensitivity
1mm
Metre ruler (0.0 - 100.0)cm 0.1mm

Vernier caliper (0.0 – 150.0)mm 0.01mm
0.01mm
Length Micrometer screw gauge (0.00 – 25.00)mm 0.01mm

Spherometer (0.00 – 7.00)mm 0.1s
0.000001s
Traveling microscope (0.00 – 220.00)mm

Time Stop watch (0.0 – 900.0)s
Timer (0.000000 – 100.000000)s

The use of Range of a measuring apparatus :
To determine the suitable apparatus can be used to measure a reading. An apparatus is suitable to measure a
reading if the range is greater than the sample.

Example :
The suitable apparatus to measure the thickness of a reference book are metre ruler, vernier calipers and
micrometer screw gauge.

The use of Sensitivity of a measuring apparatus :
To determine the best apparatus can be used to measure a reading. An apparatus is the best apparatus if it has
the smallest division of the scale.

Example :
The best apparatus to measure the thickness of a reference book is micrometer screw gauge.

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2.1 Taking reading skill :
1. Vernier caliper :

Example : 1 2 3
1
0

cm
0.01cm

Main scale = 1.1 cm

Vernier scale = 0.04 cm

2 Actual reading = 1.14 cm

2. Micrometer screw gauge :

Example : 10 15 20 Main scale = 11.0 mm
05 0
Vernier scale = 0.48 mm
mm 2
Actual reading = 11.48 mm
45

0.01mm

1

Example : Main scale = 7.5 mm
0 5 10 15 20
Vernier scale = 0.24 mm
mm 25 2
Actual reading = 7.74 mm
20

0.01mm

1

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3. Spherometer :

Example : 1 0 10
mm 7 90 20

0 80 2

0.01mm 30
70 40

-7 60
50

Main scale = 3.0 mm (upwards)

Vernier scale = 0.27 mm

Actual reading = 3.27 mm (height)

Example : 7 90
80 0
mm 70 10

0 0.01mm 2
60 20
1
50 30
-7 40

Main scale = - 4.0 mm (downwards)
Vernier scale = - 0.87 mm (opposite direction)
Actual reading = - 4.87
= 4.87 mm ( -ve = depth)

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4. Traveling microscope :

Example : .6 .7 2
.1 .2
↓ .8 .9 1.0
8 .3 .4 .5
0.01mm
cm 9

7

1

Main scale = 7.1 cm = 71.0 mm

Vernier scale = 0.41 mm

Actual reading = 71.41 mm

Example : 2

↓ .6 .7 .8 .9 1.0
.1 .2 .3 .4 .5
0.01mm
cm 89

7

1 Main scale = 7.25 cm = 72.5 mm

OR Vernier scale = 0.21 mm (lower scale)

Actual reading = 72.71 mm

Main scale = 7.20 cm = 72.0 mm

Vernier scale = 0.71 mm (upper scale)

Actual reading = 72.71 mm

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3.0 Error :

There is no such thing as a perfect measurement! All measurements have errors and uncertainties, no matter
how hard we might try to minimize them. Error and uncertainty in a measurement can arise from three possible
origins: the measuring device, the environment and the observer. Basically, there are two different types of
errors: SYSTEMATIC ERROR & RANDOM ERROR

Systematic Error

Characteristics The reading is always bias in one specific direction
Sources
(greater / less than the actual value).
Precaution
Characteristics Device Environment Observer

Sources 1. Zero error. 1. Gravity is not a 1. Reaction varies from
Precaution
2. Fault in the device constant. one person to

(metal expands when 2. Uniform air another :

the temperature resistance - reaction time.

increases). - short / long-signed.

Use perfect device. 1. Use correct value of 1. Reaction almost
gravity. constant for a
particular person
2. Create a ‘closed (carry out carefully).
system’.

Random Error

The reading is unbiased

(can be greater / can be less than the actual value).

Device Environment Observer

1. Quivering pointer 1. Changes in the 1. Parallax error.

(ammeter, voltmeter). surrounding during 2. Wrong count

experiment). (number of

oscillation).

Taking several precise readings

and then calculate the mean.

The way to overcome the zero error :

- Actual reading = (Final reading – zero error)

- Value of zero error = difference value between 0 and the initial reading.

- Sign of zero error :
→ +ve (if initial reading is greater than 0).
→ -ve (if initial reading is less than 0).

Example :

mm 0 155 2010 mm 0 155 2010
0
5
45
0
0.01mm
0.01mm
Zero error = – 0.02mm
Then, actual reading : Zero error = + 0.04mm
= (Final reading – ( – 0.02))mm Then, actual reading :
= (Final reading + 0.02))mm = (Final reading – ( + 0.04))mm
= (Final reading – 0.04))mm

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4.0 Significant Figure / Digit :

The number of significant figures in the measurement of a physical quantity :
= the number of digits before the estimated digit + the estimated digit itself.

Example :
By using vernier caliper, the diameter of a marker pen is recorded as, d = ( 2.74 + 0.01) cm.

Two digit before the
estimated digit

Estimated digit
is 4

Number of significant
figures = (2+1) = 3

The degree of precision that a measured value possesses in a number of significant figures carries by the value.
The most significant figures a value carries, the higher is the degree of precision.

There are several rules which detail how many significant figures a number has :

1. All non-zero digits are significant.
(e.g. 3.12 has 3 s.f., 45.229 has 5 s.f.)

2. Zeros between non-zero digits are significant.
(e.g. 3.012 has 4 s.f., 45.0009 has 6 s.f.)

3. Zeros beyond the decimal point at the end of a number are significant.
(e.g. 3.340 has 4 s.f.)

4. Zeros preceding the first non-zero digit are not significant. There are merely placeholders.
(e.g., 0.0034 has only 2 s.f.)

5. Digits in the exponent in exponential notation are not significant.
(e.g. 1.34 x107 has 3 s.f. (1.34) not 4)

Processing significant figures :
(a) Additional and subtraction :
When two or more measured values are added and/ or subtracted, the final calculated value must
have the same number of decimal places as that measured value which has the least number of
decimal places.
Example :
2.345 cm + 1.25 cm = 3.595 cm → 3.60 cm.

(b) Multiplication and division :
When two or more measured values are multiplied/ divided, the final calculated value must have as
many significant figures as that measured value which has the least number of significant figures.
Example :
2.345 cm X 1.25 cm = 2.93125 cm2 → 2.93 cm2

Note : Sometimes, the final answer may be obtained only after performing several intermediate calculations.
In this case results produced in intermediate calculations need not be rounded off. Round off only the final
answer. Rounded off values too early may result in greater cumulative errors.

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5.0 Uncertainty :

In an experiment, measuring a physical quantity is important but not all measured values are exactly same as the
actual values. It is because of errors we made and the apparatus we used may not in perfect conditions.
Therefore, the uncertainty of a measurement must been taken, so that the information about the accuracy of a
measurement can be obtained and it has to be recorded together with the result of the experiment.

For a quantity (best value), x with the uncertainty, ∆x, its measurement is recorded as below:
- relative uncertainty = ∆x
x
- percentage uncertainty = ∆x x100%.
x

The result should be written as (x + ∆x) ‘unit’.

Basically, there are four different types of uncertainty:
1. Uncertainty in a single reading.
2. Uncertainty in repeated readings.
3. Uncertainty in a straight line graph.
4. Uncertainty in a function.

a) Single reading :

o If the reading is taken from a single point or at the end of the scale we used :
∆x = ½ x (the smallest division from the scale).

o If the reading are taken from two points on the scale :
∆x = 2 x ½ x (the smallest division from the scale).

o If the apparatus is using the vernier scale :
∆x = 1 x (the smallest division from the scale).

Equipment Type of Smallest Example
/ apparatus reading scale of reading
Thermometer Single point 0.1oC
Metre ruler Two points 0.1cm (37.50 + 0.05)oC
Stopwatch Two points 0.1s (24.0 + 0.1)cm
Micrometer screw gauge Vernier scale 0.01mm (23.9 + 0.1)s
Vernier caliper Vernier scale 0.1mm (2.4 + 0.1)mm
Vernier scale (23.972659 + 0.000001)s
Timer 0.000001s

b) Repeated reading :

For a set of n repeated measurements, the best value is their average value.

∑n ∑n
xi |< x > −xi |
<x> = i=1 , ∆x = i=1
nn

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d) Functions :

Basically, the uncertainty for any function is represented by a formula as follow :
If the function is r = f(x,y,z,…) then the uncertainty :

∆r = dr ∆x + dr ∆y + dr ∆z

dx dy dz

Function Best Value Uncertainty
Addition / subtraction r=x+y+z ∆r = ∆x + ∆y + ∆z
Multiplication with constant k
r = kx ∆r = k∆x
Multiplication / division r = xy
∆r =  ∆x + ∆y + ∆z r
Index z x y z

r = xn ∆r = n  ∆x r

x

Example :
If the function is g = 1

2mt 2

Then, the uncertainty : g = f(m,t) …..(i)
∴ ∆g = dg ∆m + dg ∆t

dm dt

dg = d (g) dg = d (g)
dm dm
dt dt

= d  1  = d  1 2 
dm 2mt 2 dt 2mt

=  1  d (m−1) =  1  d (t −2 )
2t 2 dm
 2m  dt

=  1 (−m−2 ) =  1 (−2t −3 )
2t 2
 2m 

= -  1  …..(ii) = -  1  …..(iii)
2t 2m 2 3m
t

Substitute (ii) and (iii) into (i), we obtain :

∴ ∆g =  1 2 ∆m +  1 ∆t
2m 3m
2t t

Or ∆g =  1  ∆m + 2  1  ∆t
2t 2m m 2 2m t
t

∆g =  ∆m + 2 ∆t  1 
 m t 2t 2m

∴ ∆g =  ∆m + 2 ∆t g #
 m t 

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KOLEJ MATRIKULASI TEKNIKAL JOHOR
MINISTRY OF EDUCATION, MALAYSIA
82000 PONTIAN, JOHOR
www.kmtj.matrik.edu.my

TF015 : ENGINEERING PHYSICS
ASSIGNMENT 1

DERIVATION OF UNCERTAINTY
(GUIDELINE FOR PREPARING WORKING PROCEDURE)

THEORY :

Basically, the uncertainty for any function can be represented by a formula as follow :
“if the function is r = f(a, b, c,…) then ∆r = dr ∆a + dr ∆b + dr ∆c + ... ”

da db dc

Based on the formula, derive the uncertainty for the following functions (refers to the related physics equations) :

Experiment Related Graphical Equation Result Uncertainty
Physics
1: Equation y m xc g= ∆R =
Measurement & 1 g= 1
R = h + d2 ∆g =
uncertainty h 2gt 2 R2 0 2mt 2
2 6h y m xc µ=
2 (a) : Free fall y m xc I=
motion h = 1 gt 2 y m xc g=
2 y m xc

2 (b) : Projectile h = R2 ∆g =
motion 2gt 2
  ∆m  + 2 ∆t  g
 m   t 

5 : Static & f=µR ∆µ =
dynamics TR - τ = Iα
T = 2π l ∆I =
7 : Rotational
motion of a rigid g ∆g =

body

8 : Simple
harmonic
motion

Please refer to the manual book to identify the meaning for each symbol used.

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KOLEJ MATRIKULASI TEKNIKAL JOHOR
MINISTRY OF EDUCATION, MALAYSIA
82000 PONTIAN, JOHOR
www.kmtj.matrik.edu.my

TF025 : ENGINEERING PHYSICS
ASSIGNMENT 1

DERIVATION OF UNCERTAINTY
(GUIDELINE FOR PREPARING WORKING PROCEDURE)

THEORY :

Basically, the uncertainty for any function can be represented by a formula as follow :
“if the function is r = f(a, b, c,…) then ∆r = dr ∆a + dr ∆b + dr ∆c + ... ”
da db dc

Based on the formula, derive the uncertainty for the following functions (refers to the related physics equations) :

Experiment Related Physics Graphical Equation Result Uncertainty
1. Geometrical Equation f= 1 ∆f =  ∆m f
m 1 v -1
optics 1+1 =1 f m m
uv f ∆n =
2. Refraction y m xc ∆λ =
∆λ =
3. Newton’s Rings h - ho = Nt 1 − 1  n=
n  y m xc

d2 = 4nλR λ=
y m xc

λ=

4. Diffraction sinθ = Nnλ

y m x c N= ∆N =

5. Capacitor : Time ln Io  = t τ= ∆τ =
constant & I τ RT = ∆RT =
capacitance in y m xc
RC circuit.

6. Direct current :

Ohm’s law and V = IRT
resistance in
series & parallel. y m xc

Please refer to the manual book to identify the meaning for each symbol used.

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FLOW CHART : Constructing Data Table (Experiment in Physics)

START

Create 3 basic columns.
Column 1 : No. of Data
Column 2 : Responding Variable

(y-axis) within unit.
Column 3 : Manipulated Variable

(x-axis) within unit.

For Column 2 and 3 respectively

Is there any N Identify the related
measuring apparatus sub-equation

to measure Identify the basic
the variable? quantities

Y For every
basic quantity
Insert the uncertainty of respectively
the apparatus used

N Is the basic quantity Y
variable?

Record outside Create new column
the data table In the data table

Need to do Y Create 3 sub-columns
Repeated reading? respectively for:
Reading 1,
N Reading 2 &
Average.
Create row under column 2 & 3
to calculate Note:
A table is considered complete if it can
the coordinate of centroide point be used to :

FINISH a) plot the graph.
b) calculate the experiments result

and its uncertainty.

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