Units and Measurements Class 11 Notes:

• As human beings evolved they required different strands of comparison.
• Either something is greater or smaller, lighter or heavier is described by units and measurements.
• To live in a physical world we need to understand it, measurement is the key to that interaction and knowledge.
• Measurements are needed to know what, where, how, when, why (anything).
• Here we have provided you with complete units and measurements class 11 notes.

Units:

• Any physical quantity in a definite amount is taken as standard unit.
• The standard unit should be internationally accepted and arbitrary chosen.
• The units for the fundamental or base quantities (like length, time) are called fundamental or base units. Also, the units of all other physical quantities can be expressed as combinations of the base units.
• When such units obtained for the derived quantities, they are called derived units.

Measurement:

• The comparison of any physical quantity with a standard unit is called measurement.
• Any physical quantity is measured as how many times this quantity is with respect to standard fixed quantity (unit).

The international system of units:

Earlier there used mainly 3 system of units internationally. They are:

• CGS system: This system includes centimetre, gram and second respectively.
• FPS system: This system includes foot, pound and second respectively.
• MKS system: This system of units includes metre, kilogram and second respectively.

Now, Système Internationale d’ Unites (French name for International System of Units), is accepted internationally. It is abbreviated as SI unit.

In every field (such as scientific, technical, industrial and commercial work) we make use of international standard units (SI units).

In the international system of units, there are 7 base units. They are:

Base Quantity	Name	Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Electic Current	Ampere	A
Thermodynamic Pressure	Kelvin	K
Amount of Substance	Mole	mol
Luminous intensity	candela	cd

Along with it there are also two more supplementary fundamental units. They are used in measurement of plane angle and solid angle respectively. units and measurements class 11 notes includes both of them as:

Supplementary fundamental quantity	Supplementary unit	Symbol
Plane angle	radian	rad
Solid angle	steradian	Sr

Definition of base units:

To learn units and measurements class 11 notes it is necessary to keep in mind these terms.

1. Metre: During a time interval of (1/3lakh) of a second, the length of the path travelled by light in a vacuum is the metre.
2. Kilogram: The kilogram is equal to the mass of the platinum-iridium alloy cylinder kept at International Bureau of Weights and Measures. It’s unit has the mass of 5.0188 x 10²⁵ atoms of carbon-12.
3. Second: The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.
4. Ampere: To produce force equal to 2×10^–7 newton per metre of length between two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, the constant current required is said to be ampere.
5. Kelvin: The kelvin, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. Units and measurements class 11 notes has important topics mentioned along with it.
6. Mole: Mole is the amount of substance which contains 6.023 x 10²³ It contains as many elementary entities as there are atoms in 0.012 kilogram of carbon – 12.
7. Candela: The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×10¹² hertz and that has a radiant intensity in that direction of 1/683 watt per steradian is candela.

Some units of general use:

Some of the common terms used in this units and measurements class 11 notes are:

Name	Symbol	Value in SI unit
Minute	min	60s
Hour	h	60min = 3600 s
Day	d	24h = 86400 s
Year	y	365.25 d = 3.156 x 107s
degree	⁰	1⁰ = (π/180)rad
litre	L	1dm3 = 10-3 m3
tonne	t	103 kg
carat	c	200mg
bar	bar	0.1 MPa = 105Pa
curie	Ci	3.7 x 1010 s-1
roentgen	R	2.58 x 10-4 C/Kg
quintal	q	100kg
barn	b	100 fm2 = 10-28 m2
are	a	1dam2 = 102 m2
hectare	ha	1 hm2 = 104 m2
Standard atmospheric pressure	atm	101325 Pa = 1.013 x 105 Pa

Measurement of lengths:

• In general we have always need to measure the length in many aspects of our day to day life.
• There are several indirect methods which is used to measure beyond specific range.

Measurement of large Distances:

• To measure large distances such as distance of planets or stars from the earth a special type of method is used which is called parallax method.
• The apparent change in position of any object due to slight change in position of observation sight. This is called parallax.
• The distance between the two points of observation is called the basis.

Measurement of small distances:

• We can measure a very small size like that of a molecule (10^–8 m to 10^–10 m), by adopting some special methods.
• Even optical microscope can’t measure such range of distances as for visible light the range of wavelengths is from about 4000 Å to 7000 Å (1 angstrom = 1 Å = 10^-10 m). Hence it cannot resolve particles with sizes smaller than this.
• Electron microscopes with a resolution of 0.6 Å built which can almost resolve atoms and molecules in a material.

Dimensions of some fundamental quantities:

Fundamental quantities are assigned with their own dimensions which are used to get the dimensions of different measurement bases. Here, All the base fundamental quantities are present. When we take a dig at it, we can derive the same comprising different dimensional formula.

Derived dimensional formula of different physical quantities:

Dimensional formula of units and measurements class 11 notes is important for all exams. Please take a look at it.

Physical Quantities	Dimension	Unit
Area	L2	m2
Volume	L3	m3
Velocity	LT-1	m/s
Angular velocity	T-1	s-1
Acceleration	LT-1	m/s2
Angular Acceleration	T-2	s-2
Force	MLT-2	N(Kgm/s2)
Energy	ML2T-2	Joule(kgm2/s2)
Work	ML2T-2	Joule(kgm2/s2)
Heat	ML2T-2	Joule(kgm2/s2)
Torque	ML2T-2	Nm(kgm2/s2)
Power	ML2T-3	Watt(kgm2/s3)
Density	ML-3	N/m2(Kg/m3)
Pressure	ML-1T-2	Kg m-1/sec2
Impulse	MLT-1	Kg m/sec
Inertia	ML2	Kgm2
Luminous flux	C	Lumen(cd sr)
Illumination	CL-2	Lumen/m2(cd sr/m2)
Entropy	ML2T-2K‑1	J/degree(Kg m2/s2K)
Volume rate of flow	L3T-1	m3/sec
Kinematic viscousity	L2T-1	m2/s
Dynamic viscousity	ML-1T-1	Kg/m s
Specific weight	ML-2T-2	Kg /m2 s2

Physical quantities based on electricity:

Many physical quantities in units and measurements class 11 notes have their units and dimensions assosiated with electricity. They are:

Physical Quantities	Dimension	Unit
Electric current	QT-1	C/sec
Emf, voltage, potential	ML2T-2Q-1	Kg m2/sec2C
Resistance or  impedance	ML2T-1Q-2	Kgm2 /sec C2
Electric  conductivity	M-2L-2TQ2	sec C2/Kg m3
Capacitance	M-1L-2T2 Q2	sec2C2/Kgm2
Inductance	ML2Q-2	Kg m2 /C2
Current density	QT-1L-2	C/sec m2
Charge density	QL-3	C/m3
Magnetic flux, Magnetic induction	MT-1Q-1	Kg/sec C
Magnetic intensity	QL-1T-1	C/m sec
Magnetic vector Potential	MLT-1Q-1	Kg m/sec C
Electric field intensity	MLT-2Q-1	Kg m/sec2 C
Electric displacement	QL-2	C/m2
Permeability	MLQ-2	Kg m/C2
Permittivity,	T2Q2M-1L-3	sec2C2/Kgm3
Dielectric constant	M0L0T0	None
Frequency	T-1	sec-1
Angular frequency	T-1	sec-1
Wave length	L	M

Errors in measurement:

• The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.
• Every calculated quantity which is based on measured values, also has an error.
• To count an error the two terms that we have to keep in mind is accuracy and precision.
• The measure of the closeness of the measured value to the true value of quantity is said to be accuracy.
• The resolution or limit of the quantity to be measured is said to be precision.

Note: Errors in measurement is also an important topic in units and measurements class 11 notes. Learn it wisely.

Classification of errors in measurements:

The errors in measurement can be broadly classified as:

1. Systematic errors:

• Those errors which tend to be in one direction either in positive or in negative direction.
• An error having a non-zero mean, so that its effect is not reduced when observations are averaged are systematic errors.

The sources of systematic errors are:

• Instrumental errors: When errors arise due to imperfect design or calibration of the instrument or zero error in the instrument are instrumental errors.
• Imperfection in experimental technique or procedure: External conditions (such as changes in temperature, humidity, wind velocity, etc.) during the experiment may systematically affect the measurement.
• Personal errors: Errors which arise due to an individual’s bias, lack of proper setting of the apparatus or individual’s carelessness in taking observations without observing proper precautions, etc.

2. Random Errors:

• Errors which arise due to unpredictable fluctuations in experimental conditions are random errors.
• These errors occurs irregularly.
• They are random with respect to sign and size.

3. Least count error: The smallest value that can be measured by the measuring instrument is called its least count. The error associated with the resolution of the instrument is least count error.
4. Absolute error:

• The magnitude of the difference between the true value of the quantity and the individual measurement value is absolute error.
• It is denoted by Δa. Δa = a_mean – a_i where a_i = measured value.
• Δa_mean = \(\frac{(a_{1}\:+\: a_2\:+\:a_3\:+…+ \:a_n)}{n}\)

5. Mean Absolute Error:

• When we take the arithmetic mean of all the absolute errors is we obtain the mean absolute error of the value of the physical quantity a.
• Δa_mean = \(\frac{(|Δa1|+|Δa2 |+|Δa3|+…+ |Δan|)}{n}\)

6. Relative error:

• It is the ratio of the mean absolute error to the true value.
• Relative error = \(\frac{{Δa}_{mean}}{{a}_{mean}}\)

7. Percentage error:

• Relative error expressed in percent is said to be percentage error.
• Percentage error = \(\frac{{Δa}_{mean}}{{a}_{mean}}\) x 100

Combination of errors:

There are several mode of measurements we do. We need multiple things to combine the measurements.

• Error of a sum or a difference:
• When two quantities are added or subtracted the final absolute error is the sum of individual quantities absolute errors.
• If Z = A + B then maximum possible error in Z is ΔZ = ΔA + ΔB
• If Z = A – B then minimum possible error is in Z is ΔZ = ΔA + ΔB
• Error of a product or a quotient:
• When two quantities are multiplied or divided then the final absolute error is the sum of the relative errors in the multipliers.
• If Z = A x B or Z = \(\frac{A}{B}\) then maximum relative error in Z is \(\frac{ΔZ}{Z}\) = \(\frac{ΔA}{A}\) + \(\frac{ΔB}{B}\).
• Error in case of a measured quantity raised to a power:
• When a physical quantity raised to the power k, then relative error of that quantity is k times the relative error in the individual quantity.
• If Z = A² then \(\frac{ΔZ}{Z}\) = \(\frac{ΔA}{A}\) + \(\frac{ΔA}{A}\) = 2 \(\frac{ΔA}{A}\).
• Generally, if \(\frac{A^{p}B^{q}}{C^{r}}\)

Then, \(\frac{ΔZ}{Z}\) =p \(\frac{ΔA}{A}\) + q\(\frac{ΔB}{B}\) + r\(\frac{ΔC}{C}\)

Significant figures:

Those digits in a measured quantity which we know exactly and one additional digit that is uncertain is said to be significant figures.

There are certain rules which one keep in mind while counting of the significant figure.

• Every non- zero digits is significant figure.
• Every zero between two non-zero digits are significant figure.
• For number less than 1, the zero(s) on the right of decimal point up to the left of the first non-zero digit are not significant.
• Every zero’s to the right of a non-zero digit up to the left of a decimal point are significant.
• The terminal (right end) zero(s) in a number without a decimal point are not significant.
• The terminal (right end) zero(s) in a number with a decimal point are significant.
• The last right end zero(s) in a number with a decimal point are significant.
• When a number is greater than 1 (without decimal) the trailing zero(s) are not significant.
• Significant figures and rounding off are also important for learning units and measurements class 11 notes.

Addition and subtraction of significant figures:

• When we add or subtract, the result should be reported to the same number of decimal places as that of the number with minimum number of decimal places.
• Example: A = 25.5 gm, B = 6.92 gm then A + B = 25.5 + 6.92 = 32.42 gm. But in significant figure we write the result as 32.4 gm.

Multiplication and division in significant figure:

• When we multiply or divide, the result should be reported to the same number of significant figures as that of the number with minimum of significant figures.
• Example: If Z = 0.16 m/s x 0.152 kg = 0.02432 kg m/s but in significant figures it will be 0.02 kg m/s.

Rounding off the Uncertain Digits:

There are some rules for rounding off the uncertain digits:

1. If the digit which is to be removed is smaller than 5, then we don’t change its preceding digits. As example: 7.12 is rounded off to 7.1.
2. If the digit which is to be removed is greater than 5, then we increase the preceding digit by 1. As example: 2.57 is rounded off to 2.6.
3. When the digit which is to be removed is 5 followed by digits other than zero, then the preceding digit should be increased by 1. As example: After rounding off 4.251 we get 4.3 (1^st decimal rounding off).
4. When the digit which is to be removed is 5 or 5 followed by zero(s), then

• The preceding digit is not changed if it is even. As ex: 7.25, when rounded off, becomes 7.2.
• If the preceding digit is odd then we increase it by 1. As ex: 7.75, when rounded off, becomes 7.8

Thus each and every topic involved in units and measurements class 11 notes is important. Learn it wisely for better results.