Board exam focus — Units and Measurements (CBSE & HBSE)

Physics begins with measurement. This chapter builds the foundation for the whole syllabus by defining the International System of Units (SI), distinguishing fundamental from derived units, and developing dimensional analysis as a tool to check equations and convert units. It also introduces the unavoidable reality of measurement error and how errors combine — concepts examined in both CBSE and HBSE Class 11 papers.

Physical Quantities, SI Units and Derived Units

What is a Physical Quantity?

A physical quantity is any quantity that can be measured. Every measurement has two parts: a numerical value (magnitude) and a unit. We write a measured quantity as:

Physical quantity = numerical value x unit, i.e. Q = n u

If the unit u is made larger, the number n becomes smaller, so n u = constant. This is why 1 m of cloth and 100 cm of cloth describe the same length.

Fundamental vs Derived Units

• Fundamental (base) quantities are independent and cannot be expressed in terms of others.
• Derived quantities are obtained by combining base quantities (e.g. speed = length / time).

The Seven SI Base Units

The SI system (Systeme International d'Unites) is the internationally agreed system used throughout Physics.

There are also two supplementary units: the radian (rad) for plane angle and the steradian (sr) for solid angle.

Some Derived Units

CBSE/HBSE trap: The unit of mass is the kilogram, not the gram. Many students lose marks writing the base unit of mass as 'g'.

Measuring Very Large and Very Small Lengths

• Parallax method is used to measure the distance of nearby stars and the size of astronomical bodies.
• Convenient large units: 1 astronomical unit (AU) = 1.496 x 10^11 m; 1 light year = 9.46 x 10^15 m; 1 parsec = 3.08 x 10^16 m.
• Convenient small units: 1 angstrom = 10^-10 m; 1 fermi = 10^-15 m.

Dimensional Analysis and Checking Equations

Dimensions of a Physical Quantity

The dimensions of a physical quantity are the powers to which the base quantities are raised to represent that quantity. The three mechanical base quantities are written as [M] (mass), [L] (length) and [T] (time).

A dimensional formula expresses a quantity in terms of [M], [L], [T]. For example, the dimensional formula of force is [M L T^-2].

Dimensional Formulae of Common Quantities

The Principle of Homogeneity

Principle of homogeneity of dimensions: Every term on both sides of a physically correct equation must have the same dimensions. You cannot add quantities of different dimensions.

This principle has three main uses:

1. Checking the correctness of an equation.
2. Deriving relations among physical quantities.
3. Converting units from one system to another.

Worked Logic: Checking an Equation

Consider s = ut + (1/2) a t^2.

• LHS: [s] = [L].
• Term ut: [L T^-1][T] = [L].
• Term a t^2: [L T^-2][T^2] = [L].

All terms have dimension [L], so the equation is dimensionally consistent.

Limitations

• Dimensional analysis cannot find dimensionless constants (like 1/2 or 2*pi).
• It cannot check equations involving trigonometric, exponential or logarithmic functions directly.
• It cannot distinguish quantities with the same dimensions (e.g. work and torque are both [M L^2 T^-2]).

CBSE/HBSE trap: A dimensionally correct equation need not be physically correct, but a dimensionally incorrect equation is definitely wrong.

Significant Figures, Errors and Error Propagation

Significant Figures

The significant figures in a measurement are the reliable digits plus the first uncertain digit. Rules:

1. All non-zero digits are significant.
2. Zeros between non-zero digits are significant (e.g. 2007 has 4).
3. Leading zeros are not significant (0.0025 has 2).
4. Trailing zeros in a number with a decimal point are significant (2.300 has 4).
5. In scientific notation a x 10^b, only the digits in a count.

Types of Errors

• Systematic errors have a definite cause and direction (instrumental error, zero error, imperfect technique). They can be minimised by correction.
• Random errors occur irregularly due to unpredictable fluctuations; reduced by taking many readings and averaging.
• Least count error is the smallest value that can be measured by the instrument.

Quantifying Error

Let the true (mean) value be a_mean from n readings a_1, a_2, ..., a_n.

Combination of Errors (Error Propagation)

Sum or difference: If Z = A + B or Z = A - B, the absolute errors add: delta Z = delta A + delta B.

Product or quotient: If Z = AB or Z = A/B, the relative errors add: delta Z / Z = (delta A / A) + (delta B / B).

Power: If Z = A^p B^q / C^r, then delta Z / Z = p(delta A/A) + q(delta B/B) + r(delta C/C).

Worked Logic

For a quantity raised to a power, the fractional error is multiplied by the power. So kinetic energy E = (1/2) m v^2 has delta E / E = (delta m / m) + 2(delta v / v).

CBSE/HBSE trap: When subtracting two nearly equal numbers, the absolute errors still add — they never cancel. This greatly increases the relative error of the result.

Frequently asked questions

Are these Units and Measurements notes free?

Yes — the Units and Measurements notes for Physics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Units and Measurements notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Units and Measurements chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Units and Measurements.