JEE Physics Units and Measurements Notes Set 01

Physical Quantities, Units and Systems

Anything that can be measured is a physical quantity. A measurement states a number together with a unit, for example 5 m. Base quantities (length, mass, time, electric current, temperature, amount of substance and luminous intensity) are independent of each other, and every other quantity is derived from them.
The SI system uses the metre, kilogram, second, ampere, kelvin, mole and candela as base units. Older systems such as CGS (cm, g, s), MKS and FPS use only length, mass and time. Two supplementary quantities, plane angle (radian) and solid angle (steradian), are dimensionless.
Useful conversions: 1 nm = 10⁻⁹ m, 1 Å = 10⁻¹⁰ m, 1 fermi = 10⁻¹⁵ m, 1 kWh = 3.6 × 10⁶ J, 1 N = 10⁵ dyne, 1 J = 10⁷ erg. Light year, parsec and astronomical unit are all units of distance.

Changing the Size of Units

If a quantity has dimensions \( M^aL^bT^c \), its numerical value n changes with the units as \( n_2=n_1\left(\frac{M_1}{M_2}\right)^a\left(\frac{L_1}{L_2}\right)^b\left(\frac{T_1}{T_2}\right)^c \). A bigger unit always gives a smaller number.
When some units are scaled, express the derived unit through its formula and substitute the factors. For example, doubling the units of length, velocity and force leaves the unit of time unchanged (L/v) and quadruples the unit of energy (F × L).

Dimensions and Dimensional Analysis

The dimensional formula shows how a quantity depends on the base quantities, for example force \( [MLT^{-2}] \), energy \( [ML^2T^{-2}] \), pressure \( [ML^{-1}T^{-2}] \), power \( [ML^2T^{-3}] \) and momentum \( [MLT^{-1}] \). Pure numbers, ratios of like quantities, angles and arguments of trigonometric, exponential or logarithmic functions are dimensionless.
Principle of homogeneity: only quantities with the same dimensions can be added, subtracted or equated, so every term in a correct equation has the same dimensions. This lets us (1) check an equation, (2) convert units and (3) derive a relation between quantities as a product of powers.
Limits: it cannot give numerical constants such as ½ or 2π, it cannot handle sums of terms or trigonometric and exponential forms, and a dimensionally correct equation may still be wrong. A dimensionally wrong equation is certainly wrong.

Significant Figures

Significant figures are the digits that are known reliably plus the first uncertain digit. Rules: all non-zero digits count; zeros between non-zero digits count; leading zeros never count; trailing zeros count only after a decimal point. Writing the number in scientific form shows the count clearly: 0.01230 = 1.230 × 10⁻² has four.
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 number with the fewest. When rounding, look at the first digit to be dropped: below 5 round down, above 5 round up.
Order of magnitude: write the number as \( a\times10^n \) with 1 ≤ a < 10; the order is n if a is below about 3.16 and n + 1 otherwise.

Errors in Measurement

Absolute error is the difference between the measured and true values; relative error is the absolute error divided by the true value; percentage error is the relative error × 100. Systematic errors (zero error, faulty calibration, changed conditions) push readings in one direction and can be corrected. Random errors scatter readings both ways and are reduced by repeating the measurement. The mean absolute error is the mean of the absolute deviations from the mean.
Combination of errors (maximum error): for sums and differences, absolute errors add, \( \Delta(A\pm B)=\Delta A+\Delta B \). For products and quotients, relative errors add. For \( X=\frac{A^aB^b}{C^c} \): \( \frac{\Delta X}{X}=a\frac{\Delta A}{A}+b\frac{\Delta B}{B}+c\frac{\Delta C}{C} \), using the magnitudes of the powers. For example, in \( g=\frac{4\pi^2\ell}{T^2} \) the error in T counts twice.

Vernier Callipers and Screw Gauge

Vernier callipers: if N vernier divisions coincide with (N − 1) main scale divisions, then 1 VSD = (N − 1)/N MSD and the least count is \( \text{LC}=1\text{ MSD}-1\text{ VSD}=\frac{1\text{ MSD}}{N} \). The reading is main scale reading + (coinciding vernier division × LC).
Screw gauge: pitch is the distance moved by the screw in one full rotation, and \( \text{LC}=\frac{\text{pitch}}{\text{number of circular divisions}} \). The reading is main scale reading + (circular scale reading × LC). Always apply the zero correction: true reading = observed reading − zero error (with its sign). A reading must be quoted to the same decimal place as the least count.

Worked Examples

Worked Example 1. The period of a simple pendulum depends on its length ℓ and g. Use dimensions to find the form of T.
Answer:
Let \( T=k\,\ell^a g^b \). Then \( T=L^a(LT^{-2})^b \), so a + b = 0 and −2b = 1.
Hence \( b=-\frac12 \) and \( a=\frac12 \), giving \( T=k\sqrt{\ell/g} \). The constant k (= 2π) cannot be found by this method.

Worked Example 2. The percentage errors in the radius and mass of a solid sphere are 1% and 2%. Find the percentage error in its density.
Answer:
Density \( \rho=\frac{M}{\frac43\pi r^3} \).
\( \frac{\Delta\rho}{\rho}=\frac{\Delta M}{M}+3\frac{\Delta r}{r}=2\%+3(1\%)=5\% \).

Worked Example 3. A vernier scale has 10 divisions equal to 9 main scale divisions, each main scale division being 1 mm. Find the least count and the reading when the zero of the vernier is just past the 23 mm mark and the 4th vernier division coincides.
Answer:
LC = 1 mm − 0.9 mm = 0.1 mm.
Reading = 23 mm + 4 × 0.1 mm = 23.4 mm.