Class 11 Physics · Chapter 1 NotesUnits and Measurements

Measurement of any physical quantity means comparing it with a basic, internationally accepted reference standard called a unit. The result of a measurement is written as a number followed by a unit. Physical quantities are of two kinds: fundamental or base quantities, which cannot be expressed in terms of other quantities, and derived quantities, which are obtained by combining base quantities. The units of base quantities are called fundamental or base units, and the units of derived quantities are called derived units. A complete set of base and derived units is called a system of units. Earlier, the CGS, FPS (British) and MKS systems were widely used; today the internationally accepted system is the Système International d'Unités, abbreviated SI.

• CGS systemcentimetre, gram, second
• FPS systemfoot, pound, second
• MKS systemmetre, kilogram, second
• SI systemseven base units, decimal and convenient for conversion

In SI there are seven base quantities and their units: length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), thermodynamic temperature (kelvin, K), amount of substance (mole, mol) and luminous intensity (candela, cd). Two more units are defined for angles: the radian (rad) for plane angle, the ratio of arc length to radius, and the steradian (sr) for solid angle, the ratio of intercepted area of a sphere to the square of its radius. Both are dimensionless. Derived units are formed by combining base units; some, like the newton, joule and watt, have special names. SI prefixes such as kilo, milli and micro are used for multiples and sub-multiples, and scientific notation a × 10^b makes very large and very small numbers easy to write and compare.

• Seven base unitsm, kg, s, A, K, mol, cd
• Plane angleradian (rad); solid angle: steradian (sr); both dimensionless
• Derived units are combinations of base units
• Scientific notationa × 10^b, where 1 ≤ a < 10

Every measurement has some uncertainty, so a measured value is reported with all the digits known reliably plus the first uncertain digit. These digits are called significant figures and they indicate the precision of the measurement. The number of significant figures does not change when the unit is changed. All non-zero digits are significant; zeros between two non-zero digits are significant; zeros to the left of the first non-zero digit in a number less than 1 are not significant; trailing zeros in a number without a decimal point are not significant, but trailing zeros in a number with a decimal point are significant. Scientific notation removes ambiguity about trailing zeros.

• 2.308 cm, 0.02308 m and 23.08 mm all have four significant figures
• 4.700 m has four significant figures; 4700 m has two
• Exact numbers such as 2 in s = 2πr have infinite significant figures

In multiplication or division, the result should keep as many significant figures as the factor with the least number of significant figures. For example, dividing 4.237 g by 2.51 cm³ gives a density reported as 1.69 g cm⁻³, not the full calculator value. In addition or subtraction, the result should keep as many decimal places as the quantity with the least number of decimal places. Rounding off follows a convention: if the digit to be dropped is more than 5, the preceding digit is raised by 1; if it is less than 5, the preceding digit is left unchanged; if it is exactly 5, the preceding digit is raised by 1 when it is odd and left unchanged when it is even. In multi-step calculations, keep one extra digit in intermediate steps and round off at the end.

• Multiplication/divisionleast number of significant figures decides the result
• Addition/subtractionleast number of decimal places decides the result
• 2.745 rounded to three significant figures becomes 2.74; 2.735 becomes 2.74
• Retain one extra digit in intermediate steps to avoid rounding errors

The nature of a physical quantity is described by its dimensions, written in square brackets. The seven base quantities give the dimensions [L] for length, [M] for mass, [T] for time, [A] for electric current, [K] for thermodynamic temperature, [cd] for luminous intensity and [mol] for amount of substance. The dimensions of a quantity are the powers to which the base quantities are raised to represent it. In mechanics, quantities are usually expressed in terms of [L], [M] and [T]. For example, volume has dimensions [L³], speed [L T⁻¹], acceleration [L T⁻²] and force [M L T⁻²]. The expression showing how and which base quantities represent a quantity is its dimensional formula, and equating a quantity with its dimensional formula gives a dimensional equation.

• Volume[M⁰ L³ T⁰]
• Speed or velocity[M⁰ L T⁻¹]
• Acceleration[M⁰ L T⁻²]
• Force[M L T⁻²]
• Mass density[M L⁻³ T⁰]

Dimensional analysis rests on the principle of homogeneity: only quantities with the same dimensions can be added or subtracted, and the dimensions of all terms in a correct equation must be the same. This lets us check whether an equation is dimensionally consistent. For example, in x = x₀ + v₀t + ½at², every term on the right has the dimension of length [L], matching the left side. A dimensionally wrong equation is certainly wrong, but a dimensionally correct equation need not be the exact physical equation, because dimensionless constants cannot be found by this method. Dimensional analysis can also deduce relations among physical quantities. For a simple pendulum, assuming T = k l^x g^y m^z and equating dimensions gives T = k√(l/g), where the constant k cannot be obtained by dimensions.

• Principle of homogeneitydimensions of all terms in an equation must be equal
• Check consistencyx = x₀ + v₀t + ½at² is dimensionally correct
• Deduce relationsT = k√(l/g) for a simple pendulum
• Dimensionless constants and exact numerical factors cannot be found by dimensional analysis