JEE Physics Units and Measurements Notes Set 04

Units and the SI System

Every measurement states a number and a unit. The SI system has seven base quantities: length (metre), mass (kilogram), time (second), electric current (ampere), temperature (kelvin), luminous intensity (candela) and amount of substance (mole). Every other unit is a derived unit built from these, for example speed in m s⁻¹ and force in kg m s⁻² (newton).
Radian (plane angle) and steradian (solid angle) are supplementary units and are dimensionless.

Dimensions

The dimensions of a quantity show how it depends on the base quantities. Write them in square brackets: [M], [L], [T], [A], [K]. For example velocity is \( [LT^{-1}] \), force is \( [MLT^{-2}] \), energy is \( [ML^{2}T^{-2}] \), pressure is \( [ML^{-1}T^{-2}] \), Planck's constant is \( [ML^{2}T^{-1}] \) and the gravitational constant is \( [M^{-1}L^{3}T^{-2}] \).
Pure numbers, angles, strain, refractive index, relative density and arguments of trigonometric, logarithmic or exponential functions are dimensionless.

Principle of Homogeneity and Its Uses

1. Only quantities of the same dimensions can be added, subtracted or equated. Every term of a correct equation has the same dimensions.
2. Checking a formula: compare the dimensions of both sides.
3. Converting units: if \( n_1 u_1 = n_2 u_2 \), then \( n_2 = n_1 [M_1/M_2]^a [L_1/L_2]^b [T_1/T_2]^c \).
4. Deriving a relation: assume a power-law product of the variables and equate the powers of M, L and T. This cannot give numerical constants and cannot handle sums or trigonometric functions.

Illustration 1. Find the dimensions of \( a/b \) in \( F = a\sqrt{x} + bx^2 \).
Answer:
Each term has the dimensions of force, so \( [a] = [F][L]^{-1/2} \) and \( [b] = [F][L]^{-2} \).
\( [a/b] = [L]^{-1/2+2} = [L^{3/2}] \).

Illustration 2. Convert 1 J into a system whose units of mass, length and time are \( \alpha \) kg, \( \beta \) m and \( \gamma \) s.
Answer:
\( 1\,\text{J} = 1\,\text{kg m}^2\text{s}^{-2} \). The new units are larger, so the numerical value falls.
\( n = \alpha^{-1}\beta^{-2}\gamma^{2} = \dfrac{\gamma^2}{\alpha\beta^2} \).

Errors in Measurement

1. Absolute error is the magnitude of the difference between the measured and the true value. The mean absolute error is the average of the absolute errors of all readings.
2. Relative error is mean absolute error divided by the mean value. Percentage error is relative error multiplied by 100.
3. Systematic errors (instrument, personal, zero error) push readings one way. Random errors are reduced by taking many readings.
4. Sum or difference \( Z = A \pm B \): \( \Delta Z = \Delta A + \Delta B \).
5. Product or quotient \( Z = A^a B^b / C^c \): \( \dfrac{\Delta Z}{Z} = a\dfrac{\Delta A}{A} + b\dfrac{\Delta B}{B} + c\dfrac{\Delta C}{C} \). The maximum error always adds.

Illustration 3. A simple pendulum gives \( T = 2.0 \) s with error 0.05 s and length \( l = 1.00 \) m with error 0.01 m. Find the percentage error in \( g = 4\pi^2 l/T^2 \).
Answer:
\( \Delta g/g = \Delta l/l + 2\Delta T/T = 1\% + 2(2.5\%) = 6\% \).

Significant Figures

1. All non-zero digits are significant. Zeros between non-zero digits are significant. Leading zeros are not (0.0063 has 2). Trailing zeros after the decimal point are significant (6.3200 has 5).
2. In addition and subtraction keep as many decimal places as the least precise number. In multiplication and division keep as many significant figures as the least precise number.

Vernier Callipers and Screw Gauge

1. Least count of a vernier = 1 main scale division minus 1 vernier scale division. If n vernier divisions coincide with (n−1) main divisions of size x, least count = x/n.
2. Reading = main scale reading + (coinciding vernier division) × least count.
3. Screw gauge: least count = pitch / number of circular scale divisions. Reading = linear scale reading + circular reading × least count, then correct for zero error (subtract the zero error with its sign).