Class 11 Physics Notes

Units and Measurements Class 11 Physics Notes

Complete, exam-ready notes on units and measurements: the SI system, fundamental and derived units, dimensions and dimensional analysis, accuracy and precision, errors, and significant figures — written for CBSE, JEE and NEET revision.

Class11SubjectPhysicsCoversCBSE · JEE · NEET

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is dimensional analysis in one line?

Dimensional analysis uses the fact that physical quantities have dimensions (mass M, length L, time T) to check equation correctness, derive relations and convert units — because only quantities with the same dimensions can be added or equated.

SI Base Units and Standards

Seven base units: metre (length), kilogram (mass), second (time), ampere (current), kelvin (temperature), mole (amount), candela (luminous intensity).
Derived units are combinations of base units, e.g. newton (kg·m/s²), pascal (N/m²), joule (N·m).
Common prefixes: milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹), kilo (10³), mega (10⁶), giga (10⁹).

Dimensions and Dimensional Analysis

Dimensions

[\text{Force}] = \text{MLT}^{-2}

Every physical quantity is expressed in terms of the dimensions M, L and T. For example, force F = ma has dimensions $[F] = \text{MLT}^{-2}$ , work and energy have $[W] = \text{ML}^2\text{T}^{-2}$ .

Dimensional homogeneity: every term in a correct equation must have the same dimensions.
Deriving relations: balance the dimensions of both sides and solve for powers.
Converting units: use the dimensions as a conversion factor between unit systems.

Dimensional analysis limits

It cannot determine dimensionless constants (like π or 1/2), cannot handle trigonometric/exponential functions, and cannot distinguish quantities with the same dimensions but different meaning (e.g. torque and work are both ML²T⁻²).

Accuracy, Precision and Errors

Absolute error

\Delta a_i = |a_i - a_{\text{mean}}|

The absolute error is the deviation of a measurement from the mean. The mean absolute error is the average of individual absolute errors; the relative error is the absolute error divided by the mean value.

Systematic errors arise from instrument or method defects and are repeatable.
Random errors arise from unpredictable fluctuations; they reduce with many measurements.
Relative error:
$\frac{\Delta a}{a_{\text{mean}}}$
. Percentage error:
$\frac{\Delta a}{a_{\text{mean}}}\times100$
.

\delta a = \frac{\text{mean absolute error}}{\text{mean value}}

Relative error

Propagation of Errors

Addition/Subtraction: add the absolute errors.
Multiplication/Division: add the relative errors.
Raised to a power n: multiply the relative error by n.
Combined:
$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$
for
$Z = \frac{A}{B}$
.

\frac{\Delta Z}{Z} = \left|\frac{\Delta A}{A}\right| + \left|\frac{\Delta B}{B}\right| = \sum \text{ relative errors of factors}

Error propagation for Z = A/B

Significant Figures

All non-zero digits are significant; zeros between non-zero digits are significant.
Leading zeros are not significant; trailing zeros after a decimal point are.
Multiplication/division: result keeps the fewest significant figures of any factor.
Addition/subtraction: result keeps the fewest decimal places.
Rules for rounding: round to the nearest, and for a 5 with preceding odd digit round up.

Rounded answers

Scientific notation avoids ambiguity about significant figures: $2.40\times10^3$ clearly has three significant figures, whereas "2400" is ambiguous.

Solved Examples

Example: Two readings of a length are $5.2\,\text{cm}$ and $5.4\,\text{cm}$ . Find the absolute, relative and percentage error.

Solution: Mean $= (5.2+5.4)/2 = 5.3\,\text{cm}$ . Absolute errors are 0.1 and 0.1. Mean absolute error $= 0.1\,\text{cm}$ . Relative error $= \frac{0.1}{5.3} \approx 0.019$ ; percentage error $\approx 1.9\%$ .

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Dimensions (force)

[F] = \text{MLT}^{-2}

Relative error

\delta a = \frac{\Delta a}{a_{\text{mean}}}

Percentage error

= \frac{\Delta a}{a_{\text{mean}}}\times100

Error propagation (÷)

\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

Force [MLT⁻²], energy [ML²T⁻²], pressure [ML⁻¹T⁻²].
Add absolute errors for +/−; add relative for ×/÷.
Quantity with power n: error multiplies by n.
Dimensional analysis cannot fix dimensionless constants.
Significant figures: zeroes between digits count, leading zeroes don't.

FAQ

Common questions

What is the difference between accuracy and precision?

Accuracy is how close a measurement is to the true value; precision is how consistent repeated measurements are. A precise set can still be inaccurate (systematic error).

How do errors combine in multiplication and division?

Relative errors add: for Z = A/B, ΔZ/Z = ΔA/A + ΔB/B. For a power like Aⁿ, the error is n(ΔA/A).

What are the limitations of dimensional analysis?

It cannot determine dimensionless constants (π, 1/2), cannot handle trig/exponential functions, and cannot distinguish physically different quantities with the same dimensions.

What are the seven SI base units?

Metre, kilogram, second, ampere, kelvin, mole and candela — the fundamental units from which all other (derived) units are built.

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