Units and Measurements notes: CBSE Class 11 Physics

1. Units: the SI, derived units and practical units

A measurement is a number × unit. Base quantities (length, mass, time, electric current, temperature, amount of substance, luminous intensity) are chosen independently; every other quantity is derived.
The SI has seven base units: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd). Earlier systems: CGS (cm, g, s), FPS (ft, lb, s), MKS (m, kg, s).
Named derived units: newton $= \text{kg m s}^{-2}$ , joule $= \text{kg m}^2\text{s}^{-2}$ , watt $= \text{kg m}^2\text{s}^{-3}$ , pascal $= \text{kg m}^{-1}\text{s}^{-2}$ , hertz $= \text{s}^{-1}$ , coulomb $= \text{A s}$ .
Radian (plane angle, $\theta = s/r$ ) and steradian (solid angle, $\Omega = A/r^2$ ) were called supplementary units; they are dimensionless. A full circle is $2\pi$ rad, a full sphere $4\pi$ sr, and $1\ \text{rad} \approx 57.3^\circ$ .
Prefixes: tera $10^{12}$ , giga $10^{9}$ , mega $10^{6}$ , kilo $10^{3}$ , milli $10^{-3}$ , micro $10^{-6}$ , nano $10^{-9}$ , pico $10^{-12}$ , femto $10^{-15}$ .
Practical units: 1 light year $\approx 9.46 \times 10^{15}$ m; 1 AU $\approx 1.496 \times 10^{11}$ m; 1 parsec $\approx 3.08 \times 10^{16}$ m ( $\approx 3.26$ ly); 1 angstrom $= 10^{-10}$ m; 1 fermi $= 10^{-15}$ m.

Every tick is a factor of 100 000 in length.

A measured value has all its certain digits plus one uncertain digit: these are its significant figures.
Counting: non-zero digits count; zeros between non-zero digits count; leading zeros do not; trailing zeros count after a decimal point but not in a number without one; powers of ten do not count. Scientific form $a \times 10^{n}$ removes doubt.
Rounding: drop digit above 5, raise the last kept digit; below 5, leave it; exactly 5 with nothing after, make the last kept digit even (2.745 → 2.74, 2.735 → 2.74).
Add or subtract: keep the least number of decimal places. Multiply or divide: keep the least number of significant figures. Exact numbers (counts, defined factors) do not limit the answer.
Carry one extra digit in intermediate steps and round once at the end. Changing units does not change the number of significant figures.
Worked: $12.78 + 3.1 + 0.456 = 16.336 \to 16.3$ ; $12.4 \times 8.25 = 102.3 \to 102$ ; $482 \div 12.5 = 38.56 \to 38.6$ .

Digits 4 and 6 are certain; the 3 is an estimate.

Significant figures: The certain digits of a measurement together with the first uncertain digit.
Exact number: A counted or defined number with unlimited significant figures.

Accuracy: closeness to the true value. Precision: closeness of repeated readings to each other.
Systematic errors (zero error, faulty calibration, wrong technique, personal habit) shift readings one way; random errors scatter them and are reduced by averaging; the least count limits a single reading.
With readings $a_1 \dots a_n$ : mean $a_\text{mean}$ ; absolute error $|\Delta a_i| = |a_\text{mean} - a_i|$ ; mean absolute error $\Delta a_\text{mean} = \sum|\Delta a_i| / n$ ; relative error $= \Delta a_\text{mean}/a_\text{mean}$ ; percentage error $=$ relative error $\times 100\ \%$ .
Combining: sum or difference, $\Delta Z = \Delta A + \Delta B$ ; product or quotient, $\Delta Z / Z = \Delta A/A + \Delta B/B$ ; $Z = A^p B^q / C^r$ , $\Delta Z / Z = p\,\Delta A/A + q\,\Delta B/B + r\,\Delta C/C$ .
Report the error to one significant figure and the mean to the same decimal place. Worked: ramp times mean 2.50 s, mean absolute error 0.03 s, 1.1 %; $R = V/I = 4.80\ \Omega$ with relative error 3.67 %; $g = 4\pi^2 l/T^2 \approx 9.852$ m s $^{-2}$ with relative error 2.30 %.

Product or quotient Z = A B or A / B
$\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}$
Z = A^p B^q / C^r
$\dfrac{\Delta Z}{Z} = p\dfrac{\Delta A}{A} + q\dfrac{\Delta B}{B} + r\dfrac{\Delta C}{C}$

Mean absolute error: The average of the absolute errors of the individual readings.
Relative error: The mean absolute error divided by the mean value.

The dimensions of a quantity are the powers of the base quantities needed to express it: $[\text{M}]$ , $[\text{L}]$ , $[\text{T}]$ and so on. Velocity $[\mathrm{L}\,\mathrm{T}^{-1}]$ ; force $[\mathrm{M}\,\mathrm{L}\,\mathrm{T}^{-2}]$ ; energy $[\mathrm{M}\,\mathrm{L}^{2}\,\mathrm{T}^{-2}]$ ; pressure $[\mathrm{M}\,\mathrm{L}^{-1}\,\mathrm{T}^{-2}]$ ; $G$ $[\mathrm{M}^{-1}\,\mathrm{L}^{3}\,\mathrm{T}^{-2}]$ ; $h$ $[\mathrm{M}\,\mathrm{L}^{2}\,\mathrm{T}^{-1}]$ . Pure numbers and ratios of like quantities are dimensionless.
Principle of homogeneity: every term of a correct equation has the same dimensions; only like quantities can be added, subtracted or equated.
Uses: (1) check whether an equation can be correct; (2) deduce a relation, e.g. $T = k\,m^{a} l^{b} g^{c}$ gives $a = 0$ , $b = 0.5$ , $c = -0.5$ , so $T \propto \sqrt{l/g}$ ; $v = k F^{a}\mu^{b}$ gives $a = 0.5$ , $b = -0.5$ ; (3) convert units, $n_2 = n_1 [M_1/M_2]^{a}[L_1/L_2]^{b}[T_1/T_2]^{c}$ .
Limitations: cannot give dimensionless constants (½, 2π), cannot handle sums of terms or sin, log, exp of a dimensional quantity, and a consistent equation is not necessarily correct.

Converting a quantity of dimensions [M^a L^b T^c] between systems
$n_2 = n_1\left[\dfrac{M_1}{M_2}\right]^{a}\left[\dfrac{L_1}{L_2}\right]^{b}\left[\dfrac{T_1}{T_2}\right]^{c}$

Dimensional formula: An expression showing the powers of the base quantities that make up a quantity, such as [M L T⁻²] for force.
Principle of homogeneity: In a correct physical equation every term has the same dimensions.

Write units with every number; convert before calculating; round only at the end.
Add or subtract by decimal places; multiply or divide by significant figures.
Errors: add the relative errors in products and quotients, the absolute errors in sums and differences; a power multiplies the relative error.
To check an equation, find the dimensions of every term. Dimensions never give numerical constants.