ASSESSMENT OF TOTAL UNCERTAINTY IN THE FINAL RESULT

To assess the total uncertainty or error, it is necessary to evaluate the likely uncertainty in all the factors included in that calculation. The total uncertainty in the final result can be found as follows:

1- For Addition and subtraction:

Absolute uncertainties are added. For example, the distance “X” found by the difference between two separate position measurements.

X₁ = 10.5 ± 0.1 cm   and X₂ = 26.8 ± 0.1 cm. The difference “X” between them is recorded as

X = X₂-X₁ = (26.8 ± 0.1) – (10.5 ± 0.1)

So X = 16.3 ± 0.2 cm.

2- For Multiplication and division:

Percentage uncertainties are added.

For Example, The maximum possible uncertainty in the value of resistance R of a conductor determined from the measurements of potential difference “V” and resulting current flow ‘I’ by using   R = V/I is found as follows:

V = 5.2 ± 0.1 V

I = 0.84 ± 0.05 A

The %age uncertainty for V =0.1v/5.2v x 100/100 = about 2% 

The %age uncertainty for I = 0.05A/0.84A x 100/100 = about 6% 

Hence Total uncertainty in the value of resistance ‘R’ when ‘V’ is divided by ‘I’ is 8% The result is thus given as R = 5.2v/0.84A = 6.19A^-1 = 6.19 ohms with a %age. uncertainty of 8% because %age uncertainty for V = 2% and for I = 6% so

Total Uncertainty = º2%+6%=8%

Hence R = 6.2 ± 0.5 Ohms (Here result is rounded off to two significant figures)

3- For Power Factor

If absolute uncertainty of a measurement is known and that measurement occurs in power in a formula. Then total percentage uncertainty is calculated by multiplying the power and absolute uncertainty i.e. multiply the %age uncertainty by that power.

For Example:

For the calculation of the volume of a sphere, we use the formula V= 4/3pr³. Percentage uncertainty in volume = 3 x 5age uncertainty in radius ‘r’. When uncertainty is multiplied by power factor, then it increases the precision demand of measurement. If the radius of a small sphere is measured as 2.25 cm by vernier calipers with least count 0.01 cm, then the radius ‘r’ is recorded as

r = 2.25 ± 0.01 cm

Absolute uncertainty = least count = ± 0.01 cm

%age uncertainty in r = 0.01cm/2.25cm x 100/100 = 0.4%

Total percentage uncertainty in V = 3 x 0.4 = 1.2%

Thus volume V= 4/3pr3= 4/3 x 3.14 x (2.25)³ = 47.689 cm³ with 1.2% uncertainty

Hence the result should be recorded as V = 47.7 0.6 cm³.

PRECISION AND ACCURACY

PRECISION

Definition:

A precise measurement is the one which has less precision or absolute uncertainty. Or

An accurate measurement is the one which has less fractional or percentage uncertainty or error.

Explanation:

The precision of a measurement is determined by the instrument or device being used. The accuracy of a measurement depends upon the fractional or percentage uncertainty in that measurement.

Example:

When the length of an object is recorded as 25.5 cm by using a meter rod having smallest division in millimeter, it is the difference of two readings of the initial and final positions, The uncertainty in the single reading as described in the previous example is taken as ±0.05 cm which is now doubled (due to initial and final readings) is called “absolute uncertainty”.

Thus absolute uncertainty =±0.05±0.05 = ±0.1 cm.

The absolute uncertainty is equal to least count of the measuring instrument i.e. meter rod. This is called precision.

First Case:

Precision or absolute uncertainty (least count) = ± 0.1 cm.

As the length of the object recorded by a meter rod having least count 0.1 cm is 25.5 cm. Then fractional uncertainty = 0.1cm/25.5cm = 0.004

Percentage uncertainty = 0.1/25.5 x 100/100 = 0.4/100 = 0.4%

 

Second Case

Another measurement of length is taken by vernier caliper with least count as 0.01 cm is recorded as 0.45 cm it has Precision or absolute uncertainty (least count) = ± 0.01 cm

Fractional uncertainty = 0.01cm / 0.45cm = 0.02cm

Percentage uncertainty = 0.01/0.45 x 100/100 = 2/100 = 2.0%

This shows that the reading 25.5 cm taken by meter rod is less precise but is more accurate. In fact, it is the relative measurement, which is important. The smaller a physical quantity, the more precise instrument should be utilized. Thus the measurement 0.45 cm demands that a more precise instrument such as micrometer screw gauge with least count 0.001 cm should be utilized.

ERRORS AND UNCERTAINTIES

ERRORS AND UNCERTAINTIES

ERROR

All physical measurements are uncertain or imprecise to some extent. It is very difficult to remove all possible errors or uncertainties in a measurement. They can take place due to (i) negligence of a person (ii) inappropriate method.

There are two major types of errors which are as follows:

(1)- Random Error

(2)- Systematic Error   

(1)- Random Error

Random error is said to take place when repeated measurements of the quantity, give different values under the same conditions. It is due to some unknown reasons.

Reduction of random error:

The effect of random errors can be reduced by taking several readings of same quantity and then taking their mean (average) value. Thus average of a number of readings reduces the effect of random error.

(2)- Systematic Error

The systematic errors occur when all the measurements of a particular quantity are affected equally. These give consistent difference in the readings.

Occurrence of Systematic errors    

Systematic error can occur due to

(i)                  Zero error in measuring instruments

(ii)                Poor calibration of instruments or incorrect marking on the measuring instruments.

Reduction of systematic error:

Systematic error can be reduced by comparing the instruments with another instrument which is known to be more accurate. Thus, systematic error is reduced by applying a correction factor to all the readings taken on an instrument.

UNCERTAINTY

The uncertainty is also usually described as an error in an instrument. It can take place due to

(i)                  Inadequacy or limitation of an instrument

(ii)                Natural variations of the object being measured

(iii)               Natural defect of person’s senses.  

SCIENTIFIC NOTATION

SCIENTIFIC NOTATION

Numbers are expressed in standard form called scientific notation which uses power of ten.

The internationally accepted practice is that there should be only one non-zero digit left of decimal. For example the number 134.7 can be written as 134.7 = 1.347 x 10²

Similarly 0.0023 can be written as 0.0023 = 2.3 x 10^-3

Use of prefix:

Prefixes are used to express these large or small numbers as multiple of ten. For example

1 light year = 946 x 10¹³m = 9.46 x 10¹⁵m

Similarly for smaller number Radius of proton = 12 x 10^-16 m = 1.2 x 10^-15 m

CONVENTIONS FOR INDICATING UNITS:

Uses of SI units require special care, especially in writing prefixes. Following points should be kept in mind during the uses of units.

i)              Full name of the unit does not begin with a capital letter even if named after a scientist e.g. newton.

ii)             The symbol of unit named after a scientist has initial capital letter such as N for newton.   

iii)           The prefix should be written before the unit without any space; such as 1 x 10^-3 m is written as 1 mm. Standard prefixes are given in the following table.

Factor	Prefix	Symbol
10-18	atto	a
10-15	femto	f
10-12	pico	p
10-9	nano	n
10-6	micro	µ
10-3	milli	m
10-2	centi	c
10-1	deci	d
101	deca	da
103	kilo	k
106	mega	M
109	giga	G
1012	tera	T
1015	peta	P
1018	exa	E

iv)           A combination of base units is written each with one space apart. For example, newton meter is written as Nm.

v)            Compound prefixes are not allowed. For example, 1 µµF may be written as 1pF.

vi)           A number such as 5.0 x 10⁴ cm may be expressed in scientific notation as 5.0 x 10² m

vii)         When a multiple of a base unit is raised to a power, the power applies to the whole multiple and not the base unit alone. Thus 1km² = 1 (km)² = 1 x 10⁶ m²

viii)        Measurement in practical work should be recorded immediately in the most convenient unit e.g. micrometer screw gauge measurement in mm and the mass of the calorimeter in grams (g). But before calculation for the result, all measurements must be converted to the appropriate SI base units.    

DERIVED UNITS

DERIVED UNITS

Definition:

SI units of all other physical quantities derived from the base and supplementary units are called derived units.

For example, the unit of velocity, acceleration, force, work and momentum etc. are the derived units because the units of these quantities are the combination of two or more base units. Some of the derived units are given in the following table:

Physical Quantity	Unit	Symbol	In term of base units
Force	Newton	N	Kg ms-1
Work	joule	J	Nm = Kgm2 s-2
Power	watt	W	Js-1 = Kg m2 s-3
Pressure	Pascal	Pa	Nm-2 = Kg m-1 s-2
Charge	coulomb	C	As

Some common Examples of Derived Units

i) Speed

It is defined as the distance covered in unit time. In SI units the unit of distance is meter (m) and unit of time is second (s) therefore

Speed =  = m/s = ms^-1 Hence SI unit of speed is ms^-1

ii) Acceleration

It is defined as the change in velocity per unit time. It is denoted by “”

Acceleration = = m/s²= ms^-2

iii) Force

It is defined multiplication of mass and acceleration. It is denoted by “F”

Force = mass x acceleration = a m = m x  = kg x m/s² =  Kgms^-2

iv) Work

It is defined the distance traveled by the body under the action of force.

Work = force x distance = N-m i.e. joule

v) Momentum

It is defined the quantity of motion in a body.

Momentum = mass x velocity = Kg x m/s = Kgms^-1

SUPPLEMENTARY UNITS

SUPPLEMENTARY UNITS

The general Conference on weight and measurement has not yet get classified certain units of the SI under either base units or derived units. These SI units are called supplementary units.

For the time being this class contains only two units of purely geometrical quantities which are expressed as:

i)                    Plane Angle

ii)                   Solid Angle

(i)- RADIAN

It is the unit of plane angle and defined as the plane angle subtended by an arc at the center of the circle equal to the length of radius of the circle i.e. arc AB = r   and < AOB = 1 Radian as shown in fig.

(ii)- STERADIAN

It is the unit of solid angle and can be defined as three dimensional angle subtended at the center of a sphere by an area of its surface equal to the square of radius of the sphere as shown in fig.

INTERNATIONAL SYSTEM OF UNITS

iii) SECOND:

 Second is the base unit of time. Before 1960, the standard of time was defined in terms of the mean solar day.

1^st Definition:

It is defined as 1/86400 part of an average (mean) solar day of the year 1900 A.D. It is denoted by‘s’.

2^nd Definition:

Second is redefined as that time during which 9192631770 vibrations of cesium-133 atom take place.

iv) KELVIN:

The unit of temperature is Kelvin it can be defined as following:

Definition:

The fraction 1/273.16 of the thermodynamic temperature of the triple point of water. It is denoted by ‘K’.

Note:-

It should be noted that Triple point of a substance means the temperature at which solid, liquid and vapors phases are in equilibrium. The triple point of water is taken as 273.16 K. This definition was adopted in 1967.

v) AMPERE:

The unit of electric current is ampere. It can be defined as following:

Definition:

It is that constant current which if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section and placed a meter apart in vacuum, would produce between these conductors a force equal to 2x10^-7 newton per meter (N/m) of length. It is denoted by ‘A’. This unit of electricity was set up in 1971.

vi) CANDELA:

The unit of luminous intensity is candela. It can be defined ad following:

Definition:

The luminous in the perpendicular direction of a surface of 1/600000 square meter of a black body radiator at the solidification temperature of platinum under standard atmospheric pressure. It is denoted by ‘cd’. This definition was established in 1967.

vii) MOLE:

Mole is the unit of the amount of substance (number o particles)

Definition:

The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 Kg of carbon 12. It is denoted by ‘mol’. This unit was adopted in 1971.