1. Introduction: The Nature of Physics and Measurement | Unit and Measurement class 11 NCERT solutions

Physics is often called the “Science of Measurement.” To understand the laws of nature—whether it’s the motion of planets, the heat of the sun, or the electricity powering your phone—we need to measure quantities accurately. Lord Kelvin once said, “When you can measure what you are speaking about, and express it in numbers, you know something about it.”

1.1 What is a Physical Quantity?

A Physical Quantity is any characteristic of a material or system that can be quantified (measured) and expressed in numbers. For example, length, mass, time, temperature, and current are physical quantities. Conversely, abstract concepts like ‘love,’ ‘anger,’ or ‘beauty’ are not physical quantities because they cannot be measured on a standard scientific scale.

Physical quantities are divided into two categories:

• Fundamental (Base) Quantities: These are the building blocks of physics. They are independent of each other and cannot be defined in terms of other quantities. There are only 7 base quantities: Length, Mass, Time, Electric Current, Thermodynamic Temperature, Luminous Intensity, and Amount of Substance.
• Derived Quantities: These are quantities that are derived from the base quantities. For example, Speed is derived from Length divided by Time. Force is derived from Mass, Length, and Time.

1.2 The Measurement Process

Measurement is essentially the process of comparing an unknown physical quantity with a known standard quantity of the same nature. This standard is called a Unit.

The result of any measurement consists of two parts:

1. A Numerical Value (n): This tells us “how many times” the unit is contained in the quantity.
2. A Unit (u): The standard used for comparison.

2. Systems of Units

Historically, scientists in different parts of the world used different systems of units, which created confusion. Before the global standard was adopted, three main systems were in use:

2.1 The International System of Units (SI)

To facilitate international trade and scientific communication, the General Conference on Weights and Measures in 1971 standardized the SI System (Système International d’Unités). It is the modern metric system used universally today.

The 7 Base SI Units:

1. Length – Meter (m): Originally defined as one ten-millionth of the distance from the equator to the North Pole. Today, it is defined more precisely as the distance light travels in a vacuum in 1/299,792,458 of a second.
2. Mass – Kilogram (kg): Previously defined by a platinum-iridium cylinder stored in Paris. Since 2019, it is defined based on the Planck constant (h).
3. Time – Second (s): Defined by the vibration of Cesium-133 atoms (atomic clocks).
4. Electric Current – Ampere (A): Defined by the elementary charge of electrons.
5. Thermodynamic Temperature – Kelvin (K): Defined by the Boltzmann constant.
6. Amount of Substance – Mole (mol): One mole contains exactly 6.022 × 10^23 entities (Avogadro’s number).
7. Luminous Intensity – Candela (cd): Measures the brightness of a light source.

Supplementary Units:

In addition to the 7 base units, there are two units for measuring angles:

• Plane Angle (dθ): Measured in Radian (rad). It is the ratio of arc length to radius. (θ = arc/radius). A full circle is 2π radians.
• Solid Angle (dΩ): Measured in Steradian (sr). It is the 3D angle subtended by a surface area at the center of a sphere. (Ω = Area / radius²). A full sphere is 4π steradians.

3. Measurement Techniques: Length, Mass, and Time

Physics deals with quantities ranging from the microscopic (size of an atom ~ 10^-10 m) to the macroscopic (size of the universe ~ 10^26 m). We need different methods for different scales.

3.1 Measurement of Length

A. Direct Methods (For Normal Ranges)

• Meter Scale: Used for lengths from 1 mm to 100 m.
• Vernier Calipers: Used for lengths accurate up to 0.1 mm (or 0.01 cm). It uses a sliding scale to measure internal and external diameters.
• Screw Gauge (Micrometer): Used for measuring the thickness of wires or thin sheets, accurate up to 0.01 mm (or 0.001 cm).

B. Indirect Methods (For Large Distances)

We cannot use a measuring tape to find the distance to the moon or a star. Instead, we use the Parallax Method.

C. Indirect Methods (For Very Small Distances)

To measure the size of a molecule (like Oleic Acid), we use the Monolayer Method.

1. Prepare a dilute solution of Oleic acid in alcohol.

2. Drop a known volume of this solution onto water. The alcohol evaporates, leaving a single layer (monolayer) of acid molecules.

3. Measure the area of this film.

4. Since Volume = Area × Thickness, we can find Thickness = Volume / Area. This thickness represents the diameter of the molecule.

3.2 Measurement of Mass

Mass is the amount of matter in a body. It does not depend on temperature, pressure, or location in space.

• Common Balance: Uses a beam balance to compare an unknown mass with a known standard mass. This measures Gravitational Mass.
• Inertial Balance: Used to measure mass in zero-gravity environments by measuring the body’s resistance to change in motion (inertia).

3.3 Measurement of Time

To measure time, we need a clock. Any phenomenon that repeats itself regularly can serve as a clock (e.g., rotation of earth, pendulum). The standard for the SI second is the Cesium Atomic Clock, which is incredibly accurate. It loses or gains only 1 second in 30,000 years.

4. Accuracy, Precision, and Errors in Measurement

No measurement in science is perfectly exact. There is always some uncertainty. Understanding this uncertainty is crucial.

4.1 Accuracy vs. Precision

These terms are often confused, but they have distinct meanings in physics.

4.2 Types of Errors

Errors are the difference between the measured value and the true value.

A. Systematic Errors

These errors tend to occur in one direction (either positive or negative). They are often due to a flaw in the system.

• Instrumental Errors: Due to imperfect design or calibration (e.g., a Zero Error in Vernier Calipers where the zero marks don’t coincide).
• Imperfection in Technique: Example: Measuring body temperature by placing a thermometer under the armpit will always give a reading lower than the actual core body temperature.
• Personal Errors: Due to individual bias or carelessness, such as parallax error when reading a scale from an angle.

B. Random Errors

These occur irregularly and unpredictably due to fluctuations in conditions (like voltage change, temperature change, or vibrations). They can be positive or negative. Solution: We take the arithmetic mean (average) of a large number of readings to minimize random error.

4.3 Calculating Errors (The Math)

Let’s say we take measurements: a1, a2, a3… an.

1. True Value (Mean):
Since we often don’t know the “true” value, we assume the average of our measurements is the true value.
a_mean = (a1 + a2 + ... + an) / n

2. Absolute Error (Δa):
The magnitude of the difference between an individual measurement and the mean value.
Δa1 = |a1 - a_mean|
Note: Absolute error is always positive.

3. Mean Absolute Error (Δa_mean):
The average of all the absolute errors.
Δa_mean = (|Δa1| + |Δa2| + ... + |Δan|) / n

4. Relative Error:
The ratio of the mean absolute error to the true value.
Relative Error = Δa_mean / a_mean

5. Percentage Error:
Relative error expressed as a percentage.
Percentage Error = (Δa_mean / a_mean) × 100%

4.4 Combination (Propagation) of Errors

When we perform calculations using measured values (like adding length and width), the errors also combine. Here are the rules:

5. Significant Figures

In any measurement, the results are usually reported as a number that includes all digits that are known reliably plus the first digit that is uncertain. These digits are called Significant Figures.

5.1 Rules for Counting Significant Figures

• Non-zero digits: All non-zero digits are significant. (e.g., 1234 has 4 sig figs).
• Trapped Zeros: Zeros between two non-zero digits are significant. (e.g., 2005 has 4 sig figs).
• Leading Zeros: Zeros to the left of the first non-zero digit are NOT significant. They only indicate the position of the decimal. (e.g., 0.005 has 1 sig fig).
• Trailing Zeros: Zeros at the end of a number are significant ONLY if there is a decimal point. (e.g., 2.50 has 3 sig figs, but 2500 has only 2 sig figs).
• Exact Numbers: Constants or counting numbers (like “2” in the formula 2πr) have infinite significant figures.

5.2 Rules for Arithmetic Operations

Multiplication and Division: The final result should retain as many significant figures as there are in the original number with the least significant figures.
Example: 2.5 (2 sig figs) × 1.25 (3 sig figs) = 3.125 → Round to 3.1 (2 sig figs).

Addition and Subtraction: The final result should retain as many decimal places as there are in the number with the least decimal places.
Example: 12.52 (2 decimals) + 10.2 (1 decimal) = 22.72 → Round to 22.7 (1 decimal).

6. Dimensional Analysis

The dimensions of a physical quantity are the powers (exponents) to which the base quantities must be raised to represent that quantity. We use square brackets [ ] to represent dimensions.

Base Dimensions:

• Length: [L]
• Mass: [M]
• Time: [T]
• Current: [A]
• Temperature: [K]

6.1 Dimensional Formulae of Common Quantities

6.2 Applications of Dimensional Analysis

A. Checking the Correctness of an Equation (Principle of Homogeneity)

This principle states that only quantities with the same dimensions can be added, subtracted, or equated. If the dimensions of the Left Hand Side (LHS) equal the dimensions of the Right Hand Side (RHS), the equation is dimensionally correct.

Example: Check the equation s = ut + ½at²

• LHS: s (displacement) = [L]
• RHS Term 1: ut = Velocity × Time = [LT⁻¹][T] = [L]
• RHS Term 2: ½at² = Acceleration × Time² = [LT⁻²][T²] = [L] (½ is a constant, no dimensions)

Since [L] = [L] + [L], the equation is consistent.

B. Converting Units from One System to Another

We use the fact that n1[u1] = n2[u2].

Example: Convert 1 Joule (SI) to Ergs (CGS).

Dimension of Work/Energy = [ML²T⁻²].

a=1, b=2, c=-2.

SI System: M1=1kg, L1=1m, T1=1s.

CGS System: M2=1g, L2=1cm, T2=1s.

Formula: n2 = n1 [M1/M2]^a [L1/L2]^b [T1/T2]^c

n2 = 1 [1kg/1g]^1 [1m/1cm]^2 [1s/1s]^-2

n2 = 1 [1000]^1 [100]^2 [1]

n2 = 1000 × 10000 = 10^7.

So, 1 Joule = 10^7 Ergs.

C. Deriving Relationships Between Quantities

If we know the factors a physical quantity depends on, we can deduce the formula.

Example: Derive the formula for Centripetal Force (F) acting on a particle, knowing it depends on mass (m), velocity (v), and radius (r).

6.3 Limitations of Dimensional Analysis

• It cannot determine the value of dimensionless constants (like the ½ in equations or 2π).
• It fails if a quantity depends on more than 3 fundamental factors (because we usually have only 3 equations for M, L, T).
• It cannot derive formulas containing trigonometric (sin, cos), exponential (e^x), or logarithmic (log) functions.

7. Comprehensive Question Bank (Solved)

Test your understanding with these questions covering all topics.

Very Short Answer Questions (1 Mark)

Q1: Name the physical quantities having the dimensional formula [ML⁻¹T⁻²].
Ans: Pressure and Stress both have this dimension.

Q2: Why do we have different units for the same physical quantity?
Ans: The magnitude of a physical quantity varies over a wide range. We need different units for convenience (e.g., meters for a room, light years for stars, angstroms for atoms).

Q3: What is the number of significant figures in 0.0060?
Ans: Two (6 and 0). Leading zeros are not significant.

Short Answer Questions (2-3 Marks)

Q4: A physical quantity P is related to four observables a, b, c, and d as follows: P = (a³b²) / (√c d). The percentage errors in a, b, c, and d are 1%, 3%, 4%, and 2%, respectively. What is the percentage error in P?
Ans:
Formula for error: %Error P = 3(%a) + 2(%b) + ½(%c) + 1(%d)
Calculation: 3(1) + 2(3) + 0.5(4) + 1(2)
= 3 + 6 + 2 + 2 = 13%.

Q5: Distinguish between Dimensional Variables and Dimensionless Variables with examples.
Ans:
Dimensional Variables: Quantities that have dimensions and variable values. Examples: Force, Velocity, Area.
Dimensionless Variables: Quantities that have no dimensions but variable values. Examples: Specific Gravity, Strain, Angle.

Long Answer Questions (5 Marks)

Q6: Explain the Principle of Homogeneity of dimensions. Test the dimensional consistency of the equation for the time period of a simple pendulum: T = 2π √(l/g).
Ans:

Principle: It states that the dimensions of each term on both sides of an equation must be the same.
Testing T = 2π √(l/g):
LHS Dimension: [T]
RHS Dimension: 2π is constant. We analyze √(l/g).
Dimension of l = [L]
Dimension of g (acceleration) = [LT⁻²]
Ratio (l/g) = [L] / [LT⁻²] = [T²]
Square root of ratio = √[T²] = [T]
Since LHS [T] = RHS [T], the equation is dimensionally correct.

Q7: State the rules for determining the number of significant figures with suitable examples.
Ans:

1. All non-zero digits are significant. (Example: 156 has 3 S.F.)
2. All zeros occurring between two non-zero digits are significant. (Example: 108.05 has 5 S.F.)
3. For a number less than 1, zeros to the right of the decimal point and to the left of the first non-zero digit are not significant. (Example: 0.0045 has 2 S.F.)
4. All zeros to the right of the last non-zero digit in a number with a decimal point are significant. (Example: 2.500 has 4 S.F.)
5. In a number without a decimal point, trailing zeros are generally not significant. (Example: 1500 has 2 S.F.)