Units and Measurement

Units and Measurement

1.1 Introduction

• Measurement involves comparison with a unit.

• The result is expressed as a number with a unit.

• Physical quantities are measured directly or indirectly.

  • Fundamental quantities: independent and not expressed in terms of others (e.g., length, mass, time).

  • Derived quantities: expressed in terms of fundamental quantities (e.g., volume, velocity, force).

Fundamental and Derived Units

• Fundamental units: units for fundamental quantities.

• Derived units: combinations of base units.

1.2 The International System of Units

• A system of units includes base and derived units.

• Earlier systems: CGS (centimetre, gram, second), FPS (foot, pound, second), MKS (metre, kilogram, second).

• In 1971, the General Conference on Weights and Measures developed the Système Internationale d’ Unites (SI system).

• The SI system is used internationally in science, technology, industry, and commerce.

• The SI system has seven base units and two supplementary units.

Multiples and Submultiples of Units

(No specific content provided in the transcript, needs external knowledge to expand.)

1.3 Significant Figures

• The result of measurement includes reliable digits and the first uncertain digit.

• Significant digits: reliable digits plus the first uncertain digit.

• Example: If the period of oscillation is $1.6 s$, $1$ and $6$ are reliable, the next digit is uncertain.

Rules for Determining Significant Figures

• Rule 6: Powers of 10 in scientific notation are irrelevant to significant figures.

• Example: $4.700 m = 4.700 × 10^2 cm = 4.700 × 10^3 mm = 4.700 × 10^{-3} km$ all have 4 significant figures.

Rounding off Uncertain Digits

1. If the insignificant digit to be dropped is more than 5, the preceding digit is raised by 1.

   • Example: $2.746$ rounded to three significant figures is $2.75$ because 6 > 5.

2. If the insignificant digit to be dropped is less than 5, the preceding digit is left unchanged.

   • Example: $2.743$ rounded to three significant figures is $2.74$ because 3 < 5.

3. If the insignificant digit to be dropped is 5:

   • Case i) If the preceding digit is even, the insignificant digit is simply dropped.

     • Example: $2.745$ rounded to three significant figures is $2.74$.

   • Case ii) If the preceding digit is odd, the preceding digit is raised by 1.

     • Example: $2.735$ rounded to three significant figures is $2.74$.

Rules for Arithmetic Operations with Significant Figures

1. In multiplication or division, the final result should retain as many significant figures as the original number with the least significant figures.

   • Example: Density calculation:

     • Mass = $4.237 g$ (4 significant figures)

     • Volume = $2.51 cm^3$ (3 significant figures)

     • Density = $\frac{mass}{volume} = \frac{4.237 g}{2.51 cm^3} = 1.688047$

     • Rounded to 3 significant figures: $1.69 g/cm^3$

2. In addition or subtraction, the final result should retain as many decimal places as the number with the least decimal places.

   • Example: Sum of numbers:

     • $436.32 g$ (2 decimal places)

     • $227.2 g$ (1 decimal place)

     • $0.301 g$ (3 decimal places)

     • Sum = $663.821 g$

     • Rounded to 1 decimal place: $663.8 g$

1.4 Dimensions of Physical Quantities

• The nature of a physical quantity is described by its dimensions.

• Derived units can be expressed in terms of seven fundamental or base quantities.

• Base quantities are the seven dimensions of the physical world, denoted with square brackets [ ].

  • Length: $[L]$

  • Mass: $[M]$

  • Time: $[T]$

  • Electric Current: $[A]$

  • Thermodynamic Temperature: $[K]$

  • Luminous Intensity: $[cd]$

  • Amount of Substance: $[mol]$

• Dimensions of a physical quantity are the powers to which the base quantities are raised to represent that quantity.

• Example: Volume $[V] = length × breadth × thickness$

  • $[V] = [L][L][L] = [L^3]$

• Volume has zero dimension in mass, zero dimension in time, and three dimensions in length.

1.5 Dimensional Formulae and Dimensional Equations

• Unit of density = $kg m^{-3}$

Dimensional Formulas

1. Area = Length x Breadth

   • $[A] = [L] x [L] = [L^2]$

   • Unit of area = $m^2$

2. Volume = Length x Breadth x Height

   • $[V] = [L] x [L] x [L] = [L^3]$

   • Unit of Volume = $m^3$

3. Density = Mass/Volume

   • $[\rho]=[M]/[L^3]=[ML^{-3}]$

   • Unit of density = $kg/m^3$

4. Relative density = Density of substance/Density of water

   • $[L^0]$

5. Speed = Distance/Time

   • $[S] = [L]/[T] = [LT^{-1}]$

   • Unit of speed = $ms^{-1}$

6. Velocity = Displacement/Time

   • $[v] = [L]/[T] = [LT^{-1}]$

   • Unit of velocity = $ms^{-1}$

   • Speed and Velocity have the same dimensional formula.

7. Momentum = Mass x Velocity

   • $[p] = [M] x [LT^{-1}] = [MLT^{-1}]$

   • Unit of momentum = $kg ms^{-1}$

8. Angular Momentum = momentum x Distance

   • $[L] = [MLT^{-1}]x[L] = [ML^2T^{-1}]$

   • Unit of angular momentum = $kg m^2 s^{-1}$

9. Acceleration = Change in velocity/Time

   • $[a] = [LT^{-1}]/[T] = [LT^{-2}]$

   • Unit of acceleration = $ms^{-2}$

10. Force = Mass x Acceleration

    • $[F] = [M] x [LT^{-2}] = [MLT^{-2}]$

    • Unit of force = $kg ms^{-2}$ or newton (N)

    • $1 kg ms^{-2} = 1 N$

11. Impulse = Force x Time

    • $[I] = [MLT^{-2}] x [T] = [MLT^{-1}]$

    • Unit of Impulse = $kg ms^{-1}$

12. Work = Force x Displacement

    • $[W] = [MLT^{-2}] x [L] = [ML^2T^{-2}]$

    • Unit of work = $kg m^2 s^{-2}$ or joule (J)

    • $1 kg m^2 s^{-2} = 1 J$

13. Energy = Workdone

    • $[E] = [ML^2T^{-2}]$

    • Unit of energy = $kg m^2 s^{-2}$ or joule (J)

14. Torque = Force x perpendicular Distance

    • $[\tau] = [MLT^{-2}] x [L] = [ML^2T^{-2}]$

    • Unit of torque = $kg m^2 s^{-2}$

    • Work, Energy, and Torque have the same dimensional formula.

15. Pressure = Force/Area

    • $[P] = [MLT^{-2}]/[L^2] = [ML^{-1}T^{-2}]$

    • Unit of pressure = $kg m^{-1} s^{-2}$ or pascal (Pa)

16. Stress = Force/Area

    • $[stress] = [MLT^{-2}]/[L^2] = [ML^{-1}T^{-2}]$

    • Unit of stress = $kg m^{-1} s^{-2}$

    • Pressure and Stress have the same dimensional formula.

17. Power = Work/Time

    • $[P] = [ML^2T^{-2}]/[T] = [ML^2T^{-3}]$

    • Unit of power = $kg m^2 s^{-3}$ or watt (W)

Physical Quantities with No Dimension and No Unit

• Change in dimension divided by original dimension

• Strain $= [L]/[L] = [L^0]$

• Relative Density = Density of substance/Density of water

  • $[ML^{-3}]/[ML^{-3}] = [L^0]$

Physical Quantities Having Units, But No Dimension

• Plane angle

• Solid Angle

• Angular Displacement

Finding Dimensional Formulae

• Gravitational constant (G) using $F = G \frac{m<em>1 m</em>2}{r^2}$

  • $G = \frac{Fr^2}{m<em>1 m</em>2}$

  • $[G] = \frac{[MLT^{-2}][L^2]}{[M][M]} = [M^{-1}L^3T^{-2}]$

• Planck's constant (h) using $\lambda = \frac{h}{mv}$

  • $h = \lambda mv$

  • $[h] = [L][M][LT^{-1}] = [ML^2T^{-1}]$

1.6 Dimensional Analysis and its Applications

1. Checking the dimensional consistency (correctness) of equations.

2. Deducing relations among physical quantities.

Examples of Checking Dimensional Consistency

1. $s = ut + at$

   • $[s] = L$

   • $[ut] = LT^{-1} × T = L$

   • $[at] = LT^{-2} × T = LT^{-1}$

   • The equation is wrong because dimensions of all terms are not the same.

2. $s = ut + at^2$

   • $[s] = L$

   • $[ut] = LT^{-1} × T = L$

   • $[at^2] = LT^{-2} × T^2 = L$

   • The equation is dimensionally correct but not proved right.

3. $s = ut + \frac{1}{2}at^2$

   • [s] = L

   • $[ut] = LT^{-1} × T = L$

   • $[at^2] = LT^{-2} × T^2 = L$

   • The equation is dimensionally correct, but it is not an exact equation.

   • A dimensionally correct equation need not be an exact (correct) equation, but a dimensionally wrong (incorrect) equation must be wrong.

4. $\frac{1}{2}mv^2 = mgh$

   • $[mv^2] = M [LT^{-1}]^2 = ML^2T^{-2}$

   • $[mgh] = M LT^{-2} L = ML^2T^{-2}$

   • The equation is dimensionally correct.

5. $E = mc^2$

   • $[E] = ML^2T^{-2}$

   • $[mc^2] = M [LT^{-1}]^2 = ML^2T^{-2}$

   • The equation is dimensionally correct.

6. Given $v = x + at$, find the dimensions of $x$.

   • $[v] = [x] = [at]$

   • $[x] = [v] = LT^{-1}$

7. Given $x = a + bt + ct^2$, find the dimensions of $a, b$, and $c$ (where $x$ is in metres and $t$ in seconds).

   • $[x] = [a] = [bt] = [ct^2]$

     • $[a] = [x] = L$

     • $[b] × T = L <br>[b] = L/T = LT^{-1}$

     • $[c] × T^2 = L <br>[c] = L/T^2 = LT^{-2}$

8. Kinetic Energy Equations

   • Equations (b) and (d) are dimensionally the same and can be considered equations for kinetic energy.

9. Van der Waals Equation

   • $(P + \frac{a}{V^2})(V - b) = nRT$

   • By principle of homogeneity, the quantities with the same dimensions can be added or subtracted.

   • $[P] = [\frac{a}{V^2}]$

     • $[a] = [PV^2] = ML^{-1}T^{-2} × L^6 = ML^5T^{-2}$

   • $[b] = [V] = L^3$

Deducing Relation Among Physical Quantities

• We can deduce relation of a physical quantity which depends upto three physical quantities.

1. Kinetic Energy

   • $E \propto m^x v^y$

   • $E = k m^x v^y$

   • $ML^2 T^{-2} = M^x (LT^{-1})^y$

   • $M^1L^2 T^{-2} = M^x L^y T^{-y}$

     • $x = 1$

     • $y = 2$

     • $E = k m^1 v^2 = k mv^2$

2. Simple Pendulum

   • $T \propto m^x l^y g^z$

   • $T = k m^x l^y g^z$

   • $M^0L^0T^1 = M^x L^y (LT^{-2})^z$

   • $M^0L^0T^1 = M^x L^{y+z} T^{-2z}$

     • $x = 0$

     • $y + z = 0$

     • -2z = 1 => z = -\frac{1}{2}

     • $y = -z = \frac{1}{2}$

   • $T = k m^0 l^{\frac{1}{2}} g^{-\frac{1}{2}} = k \sqrt{\frac{l}{g}}$

   • $T = k \sqrt{\frac{l}{g}}$

Limitations of Dimensional Analysis

1. Dimensional analysis checks only the dimensional correctness of an equation, but not the exact correctness.

2. Dimensionless constants cannot be obtained by this method.

3. We cannot deduce a relation if a physical quantity depends on more than three physical quantities.

4. The method cannot derive equations involving more than one term.

5. Formulas containing trigonometric, exponential, and logarithmic functions cannot be derived.

6. It does not distinguish between physical quantities having the same dimensions.