Units and Measurement: An Introduction

Introduction to Units and Measurement

• Measurement involves comparing a physical quantity to a reference standard (unit).
• Measurement results are expressed as a number and a unit.

Fundamental and Derived Quantities

• Physical quantities can be measured directly or indirectly.
• Fundamental quantities are independent and cannot be expressed via other physical quantities (e.g., length, mass, time).
• Derived quantities can be expressed using fundamental quantities (e.g., volume, velocity, force).

Fundamental and Derived Units

• Units for fundamental quantities are fundamental or base units.
• Units for other physical quantities are combinations of base units.
• Units of derived quantities are derived units.

The International System of Units (SI)

• A system of units includes both base and derived units.
• CGS, FPS (British), and MKS systems were commonly used.
  • CGS: centimetre, gram, second.
  • FPS: foot, pound, second.
  • MKS: metre, kilogram, second.
• In 1971, the General Conference on Weights and Measures created a system with symbols, units, and abbreviations.
• The Système Internationale d' Unites (SI) is used internationally for scientific, technical, industrial, and commercial applications.
• The SI system includes seven base units and two supplementary units.

Base Quantities in the SI System

• Length is measured in metres (m).
• Mass is measured in kilograms (kg).
• Time is measured in seconds (s).
• Electric Current is measured in amperes (A).
• Thermodynamic Temperature is measured in kelvin (K).
• Amount of Substance is measured in moles (mol).
• Luminous Intensity is measured in candelas (cd).

Supplementary Quantities in the SI System

• Plane Angle is measured in radians (rad).
• Solid Angle is measured in steradians (sr).

Multiples and Submultiples of Units (examples)

• deci (d) is 10^-1.
• centi (c) is 10^-2.
• milli (m) is 10^-3.
• micro (µ) is 10^-6.
• nano (n) is 10^-9.
• pico (p) is 10^-12.
• femto (f) is 10^-15.
• deca (da) is 10.
• hecto (h) is 10^2.
• kilo (k) is 10^3.
• mega (M) is 10^6.
• giga (G) is 10^9.
• tera (T) is 10^12.
• peta (P) is 10^15.

Significant Figures

• Measurement results include reliable digits and the first uncertain digit.
• Significant digits (or figures) include reliable digits plus the first uncertain digit.

Rules for Determining Significant Figures

• Changing units does not change the number of significant figures.
• All non-zero digits are significant.
  • ex. 38-2
  • ex. 123 - 3
  • ex. 23.453 - 5
• Zeros between non-zero digits are significant, regardless of the decimal point's location.
  • ex. 1204 - 4
  • ex. 30007 - 5
  • ex. 20.03 - 4
• If a number is less than one, zeros to the right of the decimal point but to the left of the first non-zero digit are not significant.
  • ex. 0.032 - 2
  • ex. 0.004002 - 4
  • ex, 0.0132 - 3
• For numbers without a decimal point, trailing zeros are not significant.
  • ex. 4200 - 2
  • ex. 23040 - 4
  • ex. 38100 - 3
• Trailing zeros are significant in numbers with a decimal point.
  • ex. 20.0 - 3
  • ex. 43.00 - 4
  • ex. 1203.0 - 5
• Powers of 10 in scientific notation do not affect significant figures.
• All measurements (e.g., 4.700 m = 470.0 cm = 4700 mm = 0.004700 km) have four significant figures.

Rounding Off Uncertain Digits

• If the digit to be dropped is more than 5, increase the preceding digit by 1.
  • Example: 2.746 rounded to three significant figures becomes 2.75.
• If the digit to be dropped is less than 5, leave the preceding digit unchanged.
  • Example: 2.743 rounded to three significant figures becomes 2.74.
• If the digit to be dropped is 5:
  • If the preceding digit is even, drop the 5.
    • Example: 2.745 rounded to three significant figures becomes 2.74.
  • If the preceding digit is odd, increase the preceding digit by 1.
    • Example: 2.735 rounded to three significant figures becomes 2.74.

Rules for Arithmetic Operations with Significant Figures

• Multiplication/Division: The result should have as many significant figures as the original number with the least significant figures.
• Addition/Subtraction: The result should have as many decimal places as the number with the least decimal places.
• Mass is an object is 4.237g and its volume is 2.51cm3, then the density is 1.69 g/cm3

Dimensions of Physical Quantities

• The dimensions of a physical quantity describe its nature.
• Derived units can be expressed in terms of seven fundamental quantities: -[L] Length -[M] Mass -[T] Time -[A] Electric Current
  • Thermodynamic Temperature
  • Luminous intensity
  • Amount of substance
• Dimensions are the powers to which base quantities are raised.
• Volume = length × breadth × thickness.
• The dimensions of volume [V] are [L] × [L] × [L] = [L3] = [M0 L3 T0 ].
• Volume has zero dimension in mass, zero dimension in time, and three dimensions in length.

Dimensional Formulae and Dimensional Equations

• Dimensional Formula: Shows how base quantities represent the dimensions of a physical quantity.
  • Dimensional Formula of volume: [M0 L3 T0]
• Dimensional Equation: Obtained by equating a physical quantity with its dimensional formula.
  • Dimensional Equation of volume: [V] = [M0 L3 T0]

Examples of Derived Quantities and their Dimensional Formula

• Area
  • [A] = [L] x [L] = [L2] = [M0 L2 T0 ]
  • Unit: m2
• Volume
  • [V] = [L] x [L] x [L] = [L3]
  • Unit: m3
• Density
  • [ρ] = [M]/[L3] = [ML-3]
  • Unit: kg m-3
• Frequency
  • [f] = [T-1]
  • Unit: s-1
• Speed = Distance / time
  • [s] = [L]/[T] = [LT-1]
  • Unit of speed = m s-1
• Velocity = Displacement / time
  • [v] = [L]/[T] = [LT-1]
  • Unit of velocity = m s-1
• Momentum = Mass x Velocity
  • [p] = [M] x [LT-1] = [MLT-1]
  • Unit of momentum = kg m s-1
• Angular Momentum = momentum x Distance
  • [L] = [MLT-1 ] x [L]=[M L² T−1]
  • Unit of angular momentum = kg m³s-1
• Acceleration = Change in velocity / time
  • [a] = [LT-1]/[T] = [LT-2]
  • Unit of acceleration = m s-2
• Force = Mass x Acceleration
  • [F] = [M] x [LT-2]=[MLT-2 ]
  • Unit of force = kg m s-2 or newton(N), 1kgms-2 = 1N
• Impulse = Force x Time
  • [I] = [MLT-2 ] x [T] = [MLT-1 ]
  • Unit of Impulse = kg m s-1
• Work = Force x Displacement
  • [W] = [MLT-2 ]x[L]=[M L2 T−2 ]
  • Unit of work = kg m² s−2 or joule (J), 1kgm2s-2 =1J
• Energy = Workdone
  • [E] = [M L² T−2 ]
  • Unit of energy = kg m² s-2 or joule (J)
• Torque = Force x perpendicular Distance-
  • [t] = [MLT-2 ] x[L] = [M L² T−2 ]
  • Unit of torque = kgm2s-2
• Pressure = Force / Area
  • [P] = [MLT-2]/[L2] = [ML-1T-2 ]
  • Unit of pressure =kgm-1s-2 or pascal(Pa)
• Stress = Force / Area
  • [stress] = [MLT-2]/[L2] = [ML-1T-2]
  • Unit of stress = kg m¯¹ s-2
• Power = Work / Time
  • [P] = [ML2T-2]/[T] = [M L² T−3]
  • Unit of power=kg m² s−3 or watt(W)
• Change in dimension/Original dimension = [L]/[L] = [L^0]
• Density of substance /Density of water = [ML-3 ]/[ML-3 ] = [L^0 ]

Dimensional Analysis

• Checking dimensional consistency of equations.
• Deducing relation among physical quantities.
• Homogeneity principle is used to check dimensional correctness.
• This States that dimensions of each term on both sides of the equation must be same for it to be correct.
• The dimensionally correct equation need not be exact equation but dimensionally wrong (incorrect) must be wrong.

The Van der Waals Equation

• For 'n' moles of a real gas: (P + a/V^2)(V - b) = nRT

• P is the pressure, V is the volume, T is absolute temperature.

• R is the molar gas constant, and a and b are Van der Waals constants.

• Find the dimensional formula for a and b:

               [a] =[PV 2 ]
                   =ML−1 T −2 x L6
              [a] = M𝐋𝟓 𝐓 −𝟐
              [b] = [V]=[b] =𝐋𝟑