Akshat Tyagi introduces the chapter on Units and Measurements, highlighting its relevance for 11th-grade students and those preparing for NEET and JEE Main. He promises to cover NCERT and H.C. Verma (HCV) important questions, provide formula sheets, and explain concepts in detail from the basics.

The video starts by defining physics as the branch of science that explains natural phenomena and their underlying laws. It then differentiates between measurable (physical) and non-measurable (non-physical) quantities, such as length and time versus emotions like love or pain. The instructor clarifies that physics only deals with physical quantities, emphasizing that attributes like a pen's length are physical, not the pen itself.

Physical quantities are described as having both a numerical value (magnitude) and units. The historical necessity of standardized units is explained using examples like varying hand-lengths or container sizes, leading to inconsistencies. Units clarify what is being measured (e.g., centimeters for length, seconds for time) and help avoid confusion in communication.

Physical quantities are classified based on directional properties (scalars, vectors, tensors) and dependency (fundamental and derived quantities). Scalars have only magnitude (like mass, time), vectors have magnitude and direction (like force), and tensors are a complex type not focused on in this curriculum. Fundamental quantities are independent (like length, mass, time), while derived quantities depend on fundamental ones (like velocity, force).

The seven fundamental quantities are listed: Length (meter, L), Mass (kilogram, M), Time (second, T), Electric Current (ampere, A or I), Temperature (Kelvin, K or θ), Amount of Substance (mole, mol), and Luminous Intensity (candela, Cd). Derived quantities are formed by combinations of fundamental ones, such as Area (L²), Volume (L³), and Velocity (L/T).

Two supplementary quantities are introduced: plane angle and solid angle. A plane angle (θ) is formed in a 2D plane, measured in radians (rad), with the formula θ = arc length / radius. A solid angle (Ω) is a 3D generalization, measured in steradians (sr), with the formula Ω = area / radius². It's highlighted that these quantities are dimensionless but have units.

Different systems of units (FPS, CGS, MKS) existed globally before the establishment of the International System of Units (SI). SI units, derived largely from the MKS system (meter, kilogram, second), aimed to standardize measurements. The concept of derived SI units (e.g., Newton for force, Joule for work) is introduced, along with unit conversions between different systems (e.g., 1 Newton = 10⁵ dyne).

The lecture explains how to convert units using the formula n₁u₁ = n₂u₂. It also introduces light-year (distance traveled by light in a year) and astronomical unit (AU, average distance between Earth and Sun) as units of distance, often mistakenly associated with time. Definitions for 1 radian and 1 steradian are provided for school-level exams.

Dimensional analysis is presented as a powerful tool to understand physical quantities. Dimensions are defined as the powers to which fundamental quantities must be raised to represent a derived quantity (e.g., L, L², L³ for length, area, volume). Various dimensional formulas for common physical quantities like speed, acceleration, force, work, energy, and momentum are derived. The importance of understanding that different quantities can have the same dimensional formula (e.g., work, energy, torque) is emphasized.

Dimensional analysis has three main applications: converting units from one system to another, checking the correctness of physical relations, and deriving relationships between different physical quantities. The Principle of Homogeneity states that only quantities with the same dimensions can be added or subtracted. This principle is used to verify equations and determine unknown dimensions in complex formulas (e.g., Van der Waals equation).

The method of deriving formulas by relating physical quantities to fundamental dimensions (mass, length, time) raised to unknown powers (a, b, c) is demonstrated. Examples include deriving the formula for centripetal force (F = mv²/r) and the time period of a simple pendulum (T = 2π√(L/g)).

Significant figures represent the reliable digits in a measured quantity. Rules for determining significant figures are discussed: non-zero digits are always significant, zeros between non-zero digits are significant, leading zeros are not significant, and trailing zeros are significant only if a decimal point is present. The concept of rounding off based on the value of the discarded digit and the parity of the preceding digit is also covered.

Rules for performing arithmetic operations (addition, subtraction, multiplication, division) while maintaining appropriate significant figures are outlined. For addition and subtraction, the result should have the same number of decimal places as the measurement with the fewest decimal places. For multiplication and division, the result should have the same number of significant figures as the measurement with the fewest significant figures.

The concepts of accuracy (closeness to the true value) and precision (closeness of repeated measurements to each other) are introduced using an archery analogy. Different types of errors (absolute, mean absolute, relative, and percentage error) are defined and calculated based on measured and true values. The propagation of errors during addition, subtraction, multiplication, division, and powers is also discussed.

Working principles of Vernier calipers and screw gauges are explained. These instruments are used for precise measurements of length, thickness, or diameter. The video demonstrates how to read the main scale and Vernier/circular scale readings and calculate the least count. Formulas for total observed readings, including main scale reading, least count, and Vernier/circular scale reading, are provided.

The video concludes with a comprehensive summary of all topics covered, reiterating the importance of physical quantities, types of units, dimensional analysis, significant figures, errors, and measuring instruments. The instructor encourages students to practice regularly, utilize provided formula sheets, and engage with the material to build a strong foundation for future studies in physics for competitive exams.