Key Points

Physical quantities are classified as fundamental (or base) and derived. The units for fundamental quantities are called base units, and the units for derived quantities are called derived units.

The International System of Units (SI) is the modern, internationally accepted standard system. It is a decimal system, making conversions straightforward.

The SI system is based on seven fundamental units: metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity.

Besides the seven base units, there are two supplementary units: the radian (rad) for measuring plane angles and the steradian (sr) for measuring solid angles. Both are dimensionless.

Significant figures in a measured value include all the digits that are known reliably plus the first digit that is uncertain. They indicate the precision of a measurement.

All non-zero digits are significant. Zeros between non-zero digits are significant. Leading zeros are never significant. Trailing zeros are significant only if the number contains a decimal point.

In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least number of significant figures.

In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least number of decimal places.

If the digit to be dropped is greater than 5, the preceding digit is increased by 1. If it is less than 5, it is left unchanged. If it is 5, the preceding digit is made even.

The dimensions of a physical quantity are the powers to which the fundamental units (e.g., Mass [M], Length [L], Time [T]) are raised to represent that quantity.

A dimensional formula shows how a quantity is related to base quantities. A dimensional equation equates a quantity with its dimensional formula, for example, for force: $[F] = [\text{M L T}^{-2}]$.

This principle states that an equation is dimensionally correct only if the dimensions of all the terms on both sides of the equation are the same. We can only add or subtract quantities with the same dimensions.

Dimensional analysis is used to check the correctness of an equation. For example, in the equation $s = ut + \frac{1}{2}at^2$, each term has the dimension of length [L], making it dimensionally consistent.

This method can be used to derive relationships between physical quantities. For instance, the time period of a simple pendulum is found to be $T = k \sqrt{\frac{l}{g}}$, where k is a dimensionless constant.

Dimensional analysis cannot determine the value of dimensionless constants (like k, $2\pi$), cannot derive relations involving trigonometric or exponential functions, and cannot be used if a quantity depends on more than three fundamental quantities.