# UNITS AND MEASUREMENTS – PART 1

## Physical quantity

A measurable quantity is called a physical quantity. Ex: Length, mass, time, area, volume, etc.

## Fundamental quantities

The physical quantities which are independent of each other are called fundamental quantities. There are SEVEN fundamental quantities. They are Length, Mass, Time, Electric current, Thermodynamic temperature, Amount of substance, and Luminous Intensity.

## Derived quantities

The physical quantities which can be expressed in the form of a product or quotient of the fundamental quantities are called derived units. Ex: Area, Volume, Force, momentum, speed, etc.

## Unit

The basic, arbitrarily chosen, internationally accepted standard of reference which is used to express a physical quantity is called a unit.

## S. I. System

The system of units which is at present internationally accepted for measurement is the system of International (S I) and it was developed by the General conference on weights and measures in 1971. The earlier systems of units are FPS, CGS, and MKS systems.

## Fundamental units

The units used to express fundamental quantities are called Fundamental units.

This table gives the list of fundamental quantities and their units in SI.

In addition to the seven fundamental units, two supplementary units are defined which are given in the table below.

## Plane angle

It is the ratio of arc length to the radius of the circle. 𝜃 = 𝑠 𝑟 rad 𝑠 = 𝑟𝜃 Maximum plane angle around point is, 𝜃 = 2𝜋𝑟/𝑟 = 2𝜋 rad or 𝜃 = 360 degres.

Note:

(iv) 60′= 1°, 1′ = (1/60)°, 1′ = (1/60) 𝜋/180 𝑟𝑎𝑑, 1 ′ = 2.91 × 10−4 𝑟𝑎𝑑

(v) 60′′= 1′, 1′′ = (1/60)′, 1′′ = (1/60) × 2.9 × 10−4 𝑟𝑎𝑑, 1 ′′ = 4.85 × 10−6 𝑟𝑎d

## Solid angle

It is the ratio of spherical area enclosed to the square of the radius of the sphere.

𝜔 = 𝑑𝐴/𝑟2 𝑠𝑟

Maximum solid angle at the centre of the sphere is, 𝜔 = 4𝜋𝑟2/𝑟2 𝑠𝑟, 𝜔 = 4𝜋 𝑠𝑟

## Derived units

The units which can be expressed as combination of base units are called derived units.

Ex: ms-1, ms-2, kgms-1, m2, m3 etc.

## General guidelines for using symbols and units.

• Symbols for units are written in lower case starting with small letters.
• The unit names are never capitalised; however, the unit symbols are capitalised only if the symbol for a unit is derived from a proper name of scientist.
• Symbols for units do not contain any punctual marks and remain unaltered in the plural.

• It is a rational system: It uses only one unit for a given quantity.
• It is a coherent system: Every unit can be derived from seven fundamental and two supplementary units.
• It is a metric system: Multiple and sub multiples of unit can be expressed as the powers of TEN.
• It is internationally accepted.

## Measurement of length

Length of various objects or distances between the objects differ widely ranging from the radius of proton of about 10-15m to the average size of the universe with a radius of about 1026m.

Some of the simple measurement of length involves the use of

a) A metre scale for lengths from 10-3m to 102m.

b) Vernier callipers for lengths to accuracy of about 10-4m.

c) A screw gauge or spherometer to measure lengths of the order of 10-5m.

In order to measure lengths beyond these ranges some special indirect methods are adopted. One of them is the parallax method.

## Parallax

It is the change in the position of an object to its background, when the object is seen from two different positions. The distance between the two different points of observation is called the Basis. Measurement of large distance by parallax method: Let 𝐴 and 𝐵 are the two positions of observation of a distant object 𝑆. Let 𝐷 be the distance between the distant object 𝑆 and the earth. Let the distance between 𝐴 and 𝐵 be 𝑏. Let 𝜃 be the angle made by two opposite ends 𝐴 and 𝐵. As 𝐷 is very large then (𝑏/𝐷) is less than 1 and 𝐴𝐵 is taken as an arc. Then, 𝑏 = 𝐷𝜃 𝑫 = 𝒃/𝜽 Where 𝜃 is in radian and is called parallactic angle.

## Some special units of length

• 1 fermi = 1f = 10-15m
• 1 angstrom = 1Ȧ = 10-10m

(These are the shorter units of length)

• 1 astronomical unit = It is the average distance between the earth and sun. 1 AU = 1.496×1011m
• 1 light year: The distance travelled by the light in one year of time. 1ly = 9.46 ×1015m
• 1parsec: It is the distance at which an arc of length equal to one AU subtends an angle of one second at a point. 1pc = 3.08×1016m (parsec is the largest unit of length)

## Measurement of mass

Mass is the basic property of matter. It is expressed in kg, but for atomic and subatomic particles, we use unified atomic mass unit (u).

## unified atomic mass unit (u)

One unified atomic mass unit is equal to (1/12)th of the mass of an atom of carbon-12 isotope including the mass of electrons. 1u = 1.66×10-27kg.

The mass of various objects differs widely ranging from the mass of an electron about 10−30kg to the mass of universe with about 1055kg. Masses of commonly available objects are measured using a common balance. Inertial mass of an object is measured using an inertial balance. Masses of microscopic objects are determined by spectroscopic method, using a mass spectroscope. Masses of astronomical objects are estimated using Newton’s law of gravitation. Masses of binary stars are estimated using Kepler’s law of time periods.

## Measurement of time

Time measurements are done using a clock. Now we use an atomic standard of time which is based on the periodic vibrations produced in a cesium-133 atom. Caesium atomic clocks are very accurate. Measurement of time intervals ranging from 10-16s to 10-24s is estimated using photographic emulsions involved in the decay of elementary particles. Radioactive dating is used to estimate time intervals in the range of several hundred years to millions of years.

Note: A Caesium atomic clock is used at the National physical laboratory (NPL), New Delhi to maintain the Indian standard of time. Accuracy, precision of instrument and errors in measurements.

## Accuracy

The accuracy is the measure of how much close the measured value is to the true value of the quantity.

## Precision

It indicates, to what resolution or limit the quantity is measured. Least count of the instrument: The smallest value that can be measured by the measuring instrument is called least count.

Ex: least count of meter scale = 0.1 cm = 1 mm least count of vernier callipers = 0.01 cm.

## Error

The uncertainty in the measurement is called error. Errors are due to lack of accuracy and insufficient precision. The errors in measurement are classified into two types based on cause, (i) Systematic error (ii) Random error.

## Systematic error

Systematic errors are those errors that tend to be in one direction, either positive or negative and affect each measurement by same amount. these errors are due to known cause.

## Sources of systematic error (types of systematic errors)

a) Instrumental error: These errors occur due to faulty instrument or imperfect design of the measuring instrument.

b) Imperfection in experimental procedure: These errors arise due to false procedure or limitations of experimental arrangements.

c) Personal errors: These arise due to individual’s bias, lack of attentiveness or bad sights.

## Methods of reducing systematic errors

Systematic errors can be minimised by,

a) Selecting better instruments.

b) Improving experimental techniques.

c) Removing personal bias.

## Random errors

The random errors are those errors which occur irregularly due to random and unpredictable fluctuations in experimental conditions. Random errors appear due to unknown reasons.

Ex: Reading of physical balance may change due to settling of dust, change in temperature, pressure etc.

## Minimising random errors

Random errors can be minimised by repeating the measurements and taking the arithmetic mean of all measurements.

## Least count error

This error is associated with the resolution or the precision of the instrument.

## Minimising least count error

Least count error can be minimised by,

a) using instruments of higher precision.

b) improving experimental technique.

c) taking mean of all observations.

## Absolute error, relative error, and percentage error

There are three ways to express the magnitude of errors. They are,

a) Absolute error.

b) Relative error.

c) Percentage error.

## Absolute error

The difference between the individual measured value and true value is called an absolute error. The mean value am of measured values is taken as true value. If a1, a2,….an are the individual measured values in different trails then the mean value is,

The absolute error in the measured values are given by,

The absolute error may be either positive or negative.

## Mean absolute error

The arithmetic mean of the magnitude of the absolute error is called the mean absolute error (∆𝑎𝑚).

∆𝑎𝑚 = (|∆𝑎1 | + |∆𝑎2 |+. . . . . . . . +|∆𝑎𝑛|)/𝑛 = ∑|∆𝑎𝑖|/𝑛.

The result is expressed as 𝑎 = 𝑎𝑚 ± ∆𝑎𝑚. this means that the true value of 𝑎 lies between the limits 𝑎 − 𝑎𝑚 and 𝑎 + 𝑎𝑚.

## Relative error

The ratio of mean absolute error to the mean value of the quantity measured is called the relative error.

𝑹𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒆𝒓𝒓𝒐𝒓 = ∆𝒂𝒎/𝒂𝒎

## Percentage error

Relative error when expressed in percentage is called percentage error.

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟 = ∆𝑎𝑚/𝑎𝑚 × 100

𝜹𝒂 = ∆𝒂𝒎/𝒂𝒎 × 𝟏𝟎𝟎

## Combination of errors

In each measurement, there is some error and when we get the final result, these errors are combined to have the net error in the final result. Error due to a sum or difference: Let two physical quantities A and B have measured values 𝐴 ± ∆𝐴, 𝐵 ± ∆𝐵 respectively where ∆𝐴 and ∆𝐵 are their absolute error. The error ∆𝑍 in the sum 𝑍 = 𝐴 + 𝐵 is, 𝑍 ± ∆𝑍 = (𝐴 ± ∆𝐴) + (𝐵 ± ∆𝐵) = (𝐴 + 𝐵) ± (∆𝐴 + ∆𝐵). The maximum possible error in Z is, ∆𝒁 = ∆𝑨 + ∆𝑩.

For difference, 𝑍 = 𝐴 − 𝐵

𝑍 ± ∆𝑍 = (𝐴 ± ∆𝐴) − (𝐵 ± ∆𝐵)

𝑍 ± ∆𝑍 = (𝐴 − 𝐵) ± (∆𝐴 + ∆𝐵) ∆𝑍 = ∆𝐴 + ∆𝐵

The maximum error is again ∆𝒁 = ∆𝑨 + ∆𝑩.

When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities. The absolute errors due to sum or difference always add up.

## Error due to a product or a quotient

Let two physical quantities A and B have measured values 𝐴 ± ∆𝐴, 𝐵 ± ∆𝐵 respectively where ∆𝐴 and ∆𝐵 are their absolute error. The error ∆𝑍 in the product, 𝑍 = 𝐴x𝐵 is,

𝑍 ± ∆𝑍 = (𝐴 ± ∆𝐴) x (𝐵 ± ∆𝐵)

𝑍 ± ∆𝑍 = 𝐴𝐵 ± 𝐴∆𝐵 ± 𝐵∆𝐴 ± ∆𝐴∆𝐵

𝑍 ± ∆𝑍 = 𝐴𝐵 ± (𝐴∆𝐵 + 𝐵∆𝐴) (since ∆𝐴 x ∆𝐵 is small)

∆𝑍 = 𝐴∆𝐵 + 𝐵∆𝐴

Divide through by 𝐴𝐵, ∆𝑍/A𝐵 = (𝐴∆𝐵/𝐴𝐵) + (𝐵∆𝐴/𝐴𝐵)

∆𝒁/𝒁 = ∆𝑩/𝑩 + ∆𝑨/𝑨 = ∆𝑨/𝑨 + ∆𝑩/𝑩

For division,

𝑍 = 𝐴/𝐵

𝑍 ± ∆𝑍 = (𝐴 ± ∆𝐴)/(𝐵 ± ∆𝐵) = (𝐴 ± ∆𝐴)(𝐵 ± ∆𝐵)−1

𝑍 ± ∆𝑍 = 𝐴 (1 ± (∆𝐴/𝐴))𝐵−1(1 ± (∆𝐵/𝐵))−1

𝑍 ± ∆𝑍 = (𝐴/𝐵) (1 ± (∆𝐴/𝐴)) (1 ± (∆𝐵/𝐵))−1

Using binomial theorem, we have

(1 ± (∆𝐵/𝐵))−1 = 1 ± (∆𝐵/𝐵)

𝑍 ± ∆𝑍 = (𝐴/𝐵) (1 ± (∆𝐴/𝐴)) (1 ± (∆𝐵/𝐵))

𝑍 ± ∆𝑍 = 𝑍 (1 ± (∆𝐴/𝐴) ± (∆𝐵/𝐵) ± (∆𝐴∆𝐵/𝐴𝐵))

Since (∆𝐴∆𝐵/𝐴𝐵) is very small and neglecting it, we have

𝑍 ± ∆𝑍 = 𝑍 (1 ± (∆𝐴/𝐴) ± (∆𝐵/𝐵))

1 ± (∆𝑍/𝑍) = 1 ± (∆𝐴/𝐴) ± (∆𝐵/𝐵)

1 ± (∆𝑍/𝑍) = 1 ± ((∆𝐴/𝐴) + (∆𝐵/𝐵))

∆𝒁/𝒁 = (∆𝑨/𝑨)+(∆𝑩/𝑩)

When two quantities are multiplied or divided the relative error in the result is the sum of the relative errors in the individual quantities.

## Errors due to exponentiation

If 𝑍 = 𝑍2

𝑍 = 𝐴 x 𝐴

𝑍 ± ∆𝑍 = (𝐴 ± ∆𝐴) x (𝐴 ± ∆𝐴)

𝑍 ± ∆𝑍 = 𝐴𝐴 ± 𝐴∆𝐴 ± 𝐴∆𝐴 ± ∆𝐴∆𝐴 (∆𝐴∆𝐴 is very small, ∆𝐴∆𝐴 ≈ 0)

𝑍 ± ∆𝑍 = 𝐴𝐴 ± 𝐴∆𝐴 ± 𝐴∆𝐴

𝑍 + ∆𝑍 = 𝐴𝐴 ± 2𝐴∆𝐴

∆𝑍 = 2𝐴∆𝐴

∆𝑍/𝑍 = (2𝐴∆𝐴/𝐴𝐴) = 2 (∆𝐴/𝐴)

Generally, 𝑍 = 𝐴𝐾

∆𝒁/𝒁 = 𝑲 (∆𝑨/𝑨)

The relative error in a physical quantity raised to the power K is K times the relative error in the individual quantity.

This Chapter will continue in Part – 2 of Units and Measurements.

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