Units and Measurement

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Written by Lovepreet Singh Grewal

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In a physical quantity like length, mass and time, we require a standard of measurement. This standard of measurement is called the unit of that physical quantity. For example, the unit of length is metre, and a standard length of 1 metre has a precise definition. To measure the length of an object, we need to determine how many times this standard length metre is contained in the length of the object. The comparison of a physical quantity with a standard quantity is called measurement.

Table of Contents

Physical Quantities

Those quantities which can describe the laws of physics are called the physical quantity. A physical quantity is one that can be measured. Thus, length, mass, time, pressure, temperature, current and resistance are considered as physical quantities.

Classification of Physical Quantities

The physical quantities are classified into

(i) Fundamental quantities or base quantities

(ii) Derived quantities

The physical quantities that are independent of each other are called fundamental quantities. All the other quantities which can be expressed in terms of the fundamental quantities are called the derived quantities.

Units

The reference standard used to measure the physical quantities is called the unit.

Properties of Unit

  1. The unit should be of some suitable size
  2. The unit must be well-defined
  3. The unit should be easily reproducible, i.e., it should not change with place
  4. The unit must not change with time
  5. The unit should not change with physical conditions like temperature, pressure, etc.
  6. The unit must be easily comparable experimentally with similar physical quantities.

Types of Units

(i) Fundamental Units

The units defined for the fundamental quantities are called fundamental units.

(ii) Derived Units

The units of all other physical quantities which are derived from the fundamental units are called the derived units.

System of Units

(1) FPS System: In this system, the unit of length is foot, the unit of mass is pound, and the unit of time is second.

(2) CGS System: In this system, the units of length, mass and time are centimetre, gram and second, respectively.

(3) MKS System: In this system, the unit of length, mass and time are meter, kilogram and second, respectively.

(4) SI System: This system is widely used in all measurements throughout the world. The system is based on seven basic units and two supplementary units.


Basic Units

Quantity

Unit

Symbol of the Unit

Length

metre

m

Mass

kilogram

kg

Time

second

s

Temperature

kelvin

K

Electric current

ampere

A

Number of particles

mole

mol

Luminous intensity

candela

cd

Supplementary Units

Plane angle

radian

rad

Solid angle

Steradian

sr

Definition of Basic and Supplementary Units

Basic Units

1. Metre (m): One metre is the distance travelled by light in the vacuum during a time interval of (1/299792458) seconds.

2. Kilogram (kg): It is the mass of a platinum-iridium cylinder kept at the National Bureau of Weights and Measurements, Paris.

3. Second (s): The second is the time taken by the light of a specified wavelength emitted by a caesium-133 atom to execute 9192631770 vibrations.

4. Ampere (A): One ampere is that current which, when passed through two straight parallel conductors of infinite length and of negligible cross-section kept at a distance of 1 metre apart in the vacuum, produces between them a force equal to 2 x 10-7 newton per metre length.

5. Kelvin (K): It is the fraction 1/273.6 of the thermodynamic temperature of the triple point of water.

6. Candela (cd): A candela is defined as 1/60 th of the luminous intensity of 1 square centimetre of a perfect black body maintained at the freezing temperature of platinum (1773 0C).

7. Mole (md): One mole is the amount of substance that contains elementary units equal to the number of atoms in 0.012 kg of carbon-12.

Supplementary Units

1. Radian (rad): The radian is the angle subtended at the centre of the circle by the arc whose length is equal to the radius of the circle.

2. Steradian (Sr): The steradian is the solid angle subtended at the centre of a sphere by a spherical surface of an area equal to the square of its radius.

Dimensional Formula

The dimensional formula of any physical quantity is the formula that tells which of the fundamental units have been used for the measurement of that physical quantity.

How is dimensional formula written for a physical quantity?

(1) The formula of the physical quantity must be written. The quantity must be on the left-hand side of the equation.

(2) All the quantities on the right-hand side of the formula must be written in terms of fundamental quantities like mass, length and time.

(3) Replace mass, length and time with M, L and T, respectively.

(4) Write the powers of the terms.

Characteristics of Dimensions

(1) Dimensions do not depend on the system of units.

(2) Quantities with similar dimensions can be added or subtracted from each other.

(3) Dimensions can be obtained from the units of the physical quantities and vice versa.

(4) Two different quantities can have the same dimension.

(5) When two dimensions are multiplied or divided, it will form the dimension of the third quantity.

Dimensional Analysis

The dimensional formula can be used to

(1) Check the correctness of the equation.

(2) Convert the unit of the physical quantity from one system to another.

(3) Deduce the relation connecting the physical quantities.

Units and Dimensions of a Few Derived Quantities

Physical Quantity

Unit

Dimensional Formula

Displacement

m

M0L1T0

Area

m2

M0L2T0

Volume

m3

M0L3T0

Velocity

ms-1

M0L1T-1

Acceleration

ms-2

M0L1T-2

Density

Kg m-3

M1L-3T0

Momentum

Kg ms-1

M1L1T-1

Work/Energy/Heat

Joule (or) Kg m2/sec2

M1L2T-2

Power

Watt (W) (or) Joule/sec

M1L2T-3

Angular Velocity

rad s-1

M0L0T-1

Angular Acceleration

rad s-2

M0L0T-2

Moment of Inertia

Kg m2

M1L2T0

Force

Newton (or) Kg m/sec2

M1L1T-2

Pressure

Newton/m (or) Kg m-1/sec2

M1L-1T-2

Impulse

Newton sec (or) Kg m/sec

M1L1T-1

Inertia

Kg m2

M1L2T0

Electric Current

Ampere (or) C/sec

QT-1

Resistance/Impedance

Ohm (or) Kg m2/sec C2

ML2T-1Q-2

EMF/Voltage/Potential

Volt (or) Kg m2/sec2 C

ML2T-2Q-1

Permeability

henry/m (or) Kg m/C2

MLQ-2

Permittivity

Farad/m (or) sec2C2/Kgm3

T2Q2M-1L-3

Frequency

Hertz (or) sec-1

T-1

Wavelength

m

L1

Principle of Homogeneity

According to the principle of homogeneity of dimensions, all the terms in a given physical equation must be the same.

Ex. s = ut + (½) at2

Dimensionally

[L] = [LT-1.T] + [LT-2. T2] [L] = [L] + [L]

Defects of Dimensional Analysis

  1. While deriving the formula, the proportionality constant cannot be found.
  2. The equation of a physical quantity that depends on more than three independent physical quantities cannot be deduced.
  3. This method cannot be used if the physical quantity depends on more parameters than the number of fundamental quantities.
  4. The equations containing trigonometric functions and exponential functions cannot be derived

Points to Remember

  • Those quantities which can describe the laws of physics are called the physical quantity. Example: length, mass and time.
  • Physical quantities can be classified as fundamental quantities and derived quantities.
  • The reference standard used to measure the physical quantities is called the unit. Units are classified as fundamental units and derived units.
  • The SI system is the most commonly used system of units
  • The SI is based on seven basic units and two supplementary units.
  • The dimensional formula of any physical quantity is the formula that tells which of the fundamental units have been used for the measurement of that physical quantity.
  • The dimensional formula follows the principle of homogeneity

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                                                            MCQ                                                                                            

1. The symbol to represent “Amount of Substance” is ________

  1. K
  2. A
  3. Cd
  4. mol

Answer: (d) mol

Explanation: The symbol to represent Amount of Substance is mol.

2. Which among the following is the Supplementary Unit——–

  1. Mass
  2. Time
  3. Solid angle
  4. Luminosity

Answer: (c) Solid angle

Explanation: Supplementary units are plane angles and solid angles. Other units mentioned are base units.

3. What is the unit of solid angle?

  1. second
  2. Steradian
  3. kilogram
  4. candela

Answer: (b) Steradian

Explanation: Steradian is the unit of solid angle.

4. AU is the unit of ________

  1. Astronomy Unit
  2. Astronomical unit
  3. Astrological Unit
  4. Archaeological Unit

Answer: (b) Astronomical unit

Explanation: Astronomical unit is the average distance of the Sun from the Earth. It is represented by the symbol AU.

5. Dimensions of kinetic energy is the same as that of ________

  1. Acceleration
  2. Velocity
  3. Work
  4. Force

Answer: (c) Work

Explanation: Dimensions of kinetic energy and work are the same.

6. Farad is the unit of ________

  1. Luminosity
  2. Wavelength
  3. Permittivity
  4. Inertia

Answer: (c) Permittivity

Explanation: Permittivity is the unit of Farad.

7. Electron volt is a unit of

  1. Luminosity
  2. Frequency
  3. Force
  4. Energy

Answer: (d) Energy

Explanation: One of the units of energy is electron volt.

8. Joule second is the unit of

  1. Force
  2. Angular momentum
  3. Energy
  4. Power

Answer: (b) Angular momentum

Explanation: Angular momentum is also known as rotational momentum. The total angular momentum of a closed system remains constant.

9. The smallest value which is measured using an instrument is known as ________

  1. Absolute count
  2. Least count
  3. Round off value
  4. Minimum count

Answer: (b) Least count

10. Which is the system of unit

  1. SMS system
  2. MKP system
  3. FPS System
  4. CJS System

Answer: (c) FPS System

Explanation: FPS System is one of the systems of units. Foot, Pound, and Second (FPS system).

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