Units and Measurements Class 11 Physics Notes

Key Definitions

Physical Quantity

A quantity that can be measured and expressed in terms of magnitude and unit, e.g., length, mass, time.

Unit

A standard amount of a physical quantity, chosen to measure the quantity of the same kind, e.g., metre for length, kilogram for mass.

Base Quantities (Fundamental Quantities)

Physical quantities that are independent of each other and cannot be expressed in terms of other physical quantities, e.g., length, mass, time, electric current, thermodynamic temperature, amount of substance, luminous intensity.

Derived Quantities

Physical quantities whose units are expressed in terms of the base units, e.g., velocity (length/time), force (mass x acceleration).

Dimension

The power to which the fundamental units are raised to represent a derived unit of a physical quantity, represented by square brackets, e.g., [M], [L], [T].

Accuracy

The degree of closeness of a measured value to the true value of the quantity.

Precision

The resolution or limit to which a quantity is measured; refers to the closeness of two or more measurements to each other.

Significant Figures

The reliable digits plus the first uncertain digit in a measured value.

Systems of Units and the SI System

Historically, various systems of units were used, leading to confusion and inconsistencies in scientific and commercial exchanges. The most common ones included the CGS (Centimetre-Gram-Second) system, the FPS (Foot-Pound-Second) system, and the MKS (Metre-Kilogram-Second) system. Each had its advantages but lacked universal acceptance, hindering international collaboration.

To overcome these issues, the International System of Units (SI) was adopted in 1960. SI is a coherent and rationalised system, widely accepted globally, and forms the basis of all modern scientific and technological work. It is based on seven fundamental (base) units and two supplementary units. The base units are for: 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). The two supplementary units are the radian (rad) for plane angles and the steradian (sr) for solid angles. All other physical quantities, known as derived quantities, have units that can be expressed as algebraic combinations of these base units. For example, the unit of force, Newton (N), is derived from kg m/s².

Understanding the SI system is paramount as it ensures consistency in physical calculations and communication worldwide. It simplifies the teaching and learning of physics and prevents errors that could arise from unit conversions. Always ensure your final answers in physics problems are expressed in appropriate SI units unless specified otherwise.

Dimensional Analysis

Dimensional Analysis Example

• {"title":"Check dimensional consistency of E = mc²","bodyMarkdown":"Dimensions of E (Energy) = [M¹L²T⁻²]\nDimensions of m (mass) = [M¹]\nDimensions of c (speed of light) = [L¹T⁻¹]\nDimensions of mc² = [M¹] ([L¹T⁻¹])² = [M¹L²T⁻²]\nSince Dimensions of E = Dimensions of mc², the equation is dimensionally consistent."}
• {"title":"Dimensions of Planck's constant (h)","bodyMarkdown":"From E = hν (where E is energy, ν is frequency)\nh = E/ν\nDimensions of E = [M¹L²T⁻²]\nDimensions of ν = [T⁻¹]\nDimensions of h = [M¹L²T⁻²] / [T⁻¹] = [M¹L²T⁻¹]"}

Errors in Measurement & Significant Figures

Every measurement involves some degree of uncertainty or error. Understanding and quantifying these errors is crucial for assessing the reliability of experimental data. Errors can be broadly classified into systematic errors and random errors.

Systematic errors are those that tend to be in one direction (either positive or negative) and arise due to: instrumental errors (faulty calibration), imperfection in experimental technique or procedure, and personal errors (due to individual's bias). These errors can often be minimised by improving the experimental setup or technique.

Random errors are those that occur irregularly and are thus random in magnitude and direction. They arise due to unpredictable fluctuations in experimental conditions (e.g., temperature, voltage supply, mechanical vibrations) or personal least count error. These errors cannot be eliminated completely but can be reduced by taking a large number of observations and calculating the arithmetic mean.

Absolute error is the magnitude of the difference between the individual measurement and the true value (or mean value). Mean absolute error is the arithmetic mean of all the absolute errors. Relative error is the ratio of the mean absolute error to the mean value of the quantity measured. Percentage error is the relative error expressed as a percentage.

Combination of Errors:

• In addition/subtraction: The absolute errors add up.

If Z = A + B or Z = A - B, then ΔZ = ΔA + ΔB

• In multiplication/division: The fractional (relative) errors add up.

If Z = A x B or Z = A / B, then ΔZ/Z = ΔA/A + ΔB/B

• In powers: If Z = Aⁿ, then ΔZ/Z = n(ΔA/A)

Significant figures are the digits in a number that carry meaningful information, contributing to its precision. They include all the certain digits plus the first uncertain digit. The number of significant figures reflects the precision of a measurement. Rules for determining significant figures are:

1. All non-zero digits are significant. (e.g., 2345 has 4 significant figures).
2. All zeros between two non-zero digits are significant. (e.g., 2005 has 4 significant figures).
3. Leading zeros (zeros before non-zero digits) are not significant. (e.g., 0.0023 has 2 significant figures).
4. Trailing zeros (zeros at the end of a number) are significant ONLY if the number contains a decimal point. (e.g., 2.000 has 4 significant figures; 2000 has 1 significant figure).
5. Exact numbers or defined quantities have an infinite number of significant figures. (e.g., the number of sides in a triangle is 3, exactly).

Rules for arithmetic operations with significant figures:

• Addition/Subtraction: The result should have no more decimal places than the measurement with the fewest decimal places.
• Multiplication/Division: The result should have no more significant figures than the measurement with the fewest significant figures.

Significant Figures Example

• {"title":"Identify significant figures:","bodyMarkdown":"a) 3456 -> 4 sig figs\nb) 3004 -> 4 sig figs\nc) 0.0078 -> 2 sig figs\nd) 5.00 -> 3 sig figs\ne) 5000 -> 1 sig fig (no decimal point)"}
• {"title":"Arithmetic with sig figs:","bodyMarkdown":"a) 2.3 + 1.25 = 3.55 -> round to 3.6 (one decimal place)\nb) 2.3 x 1.25 = 2.875 -> round to 2.9 (two sig figs)"}

Must Remember

• SI units are fundamental for consistency in physics. Always use them unless specified.
• Dimensional analysis is for checking consistency, not correctness of numerical constants.
• An equation can be dimensionally correct but physically wrong (e.g., E = 2mc²).
• Errors are inherent in all measurements; minimise systematic errors, reduce random errors by averaging.
• Accuracy refers to closeness to true value, precision refers to resolution of measurement.
• Always apply significant figure rules in final calculations to reflect the precision of your measurements.
• Parallax error is a common personal error in reading scales; always view perpendicularly.

Exam Tip: Avoiding Common Traps

Many students lose marks in Units and Measurements by ignoring unit conversions or applying significant figure rules incorrectly. Always double-check your units before calculations, especially when combining different physical quantities. For dimensional analysis, be meticulous with powers and remember that trigonometric functions, logarithms, and exponentials are dimensionless. When dealing with error propagation, clearly distinguish between absolute and relative errors. Pay close attention to trailing zeros with or without decimal points when counting significant figures, as this is a frequent trap! Practice unit conversions and error calculations regularly.

Practice Questions with Solutions

• Q: What is the difference between a base quantity and a derived quantity? A: A base quantity is independent and cannot be expressed in terms of other quantities (e.g., length). A derived quantity is expressed in terms of base quantities (e.g., speed, which is length/time).
• Q: Name the seven base units of the SI system. A: Metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd).
• Q: What are the dimensions of gravitational constant 'G'? A: From F = Gm₁m₂/r², G = Fr²/m₁m₂. Dimensions of G = [M¹L¹T⁻²][L²]/[M²] = [M⁻¹L³T⁻²].
• Q: How many significant figures are there in 0.05040? A: There are 4 significant figures. Leading zeros are not significant, but zeros between non-zero digits and trailing zeros after a decimal point are significant.

Frequently Asked Questions