Chapter 1: Units and Measurement Long Questions | physics Class 11 CBSE

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Chapter 1: Units and Measurement Long Questions | physics Class 11 CBSE | ByteNotes.in

Long Answer

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Medium

Question 1

Long Form

Describe the SI system of units. Why was it developed, and what are its key features that make it superior to earlier systems?

The SI (Système Internationale d’Unités) was developed by the BIPM in 1971 and revised in November 2018, replacing the earlier CGS, FPS, and MKS systems that caused inconsistency across countries.
It is founded on seven base units — metre, kilogram, second, ampere, kelvin, mole, and candela — each defined precisely using fixed values of universal physical constants.
Its decimal structure makes conversion between multiples (using SI prefixes) straightforward, unlike the FPS system where 12 inches make a foot and 3 feet a yard.
Derived units are formed coherently from the seven base units; some receive special names (e.g., newton, joule, watt), and all carry internationally agreed symbols.
The 2018 revision redefined all base units in terms of fixed numerical values of constants such as the Planck constant h and the elementary charge e, eliminating reliance on physical artefacts.
The SI is applicable across all scientific, technical, industrial, and commercial fields worldwide, ensuring seamless communication and reproducibility of measurements.

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Question 2

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Explain the concept of significant figures. Discuss the rules for determining significant figures and the arithmetic rules for operations involving them.

Significant figures are the reliable digits plus the first uncertain digit in a measured value; they indicate the precision of a measurement, which depends on the least count of the instrument.
Key rules for counting: all non-zero digits are significant; zeros between non-zero digits are significant; leading zeros are not significant; trailing zeros with a decimal point are significant; trailing zeros without a decimal point are not.
Scientific notation eliminates all ambiguity: every digit in the base number a of a × 10ᵇ is significant, and the power of 10 has no bearing on the count.
For multiplication or division, the result is rounded to as many significant figures as the least precise factor; for example, density = 4.237/2.51 = 1.69 g cm⁻³ (three sig. figs.).
For addition or subtraction, the result is rounded to the same number of decimal places as the term with the fewest decimal places; for example, 436.32 + 227.2 + 0.301 = 663.8 g.
In multi-step calculations, one extra significant figure should be retained in intermediate steps to prevent accumulation of rounding errors, with final rounding only at the end.

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Question 3

Long Form

Describe the concept of dimensions of physical quantities. Derive the dimensional formulae for velocity, force, and mass density, and write their dimensional equations.

The dimensions of a physical quantity are the powers to which the seven base quantities (L, M, T, A, K, cd, mol) must be raised to represent it; they describe the physical nature of the quantity.
Velocity = displacement/time, so its dimensions are [L]/[T] = [M⁰ L T⁻¹]; the dimensional equation is [v] = [M⁰ L T⁻¹].
Force = mass × acceleration = mass × length/time², giving dimensions [M][L T⁻²] = [M L T⁻²]; the dimensional equation is [F] = [M L T⁻²].
Mass density = mass/volume = mass/length³, giving dimensions [M]/[L³] = [M L⁻³ T⁰]; the dimensional equation is [ρ] = [M L⁻³ T⁰].
In these representations the magnitudes are irrelevant; all forms of velocity (average, instantaneous, relative) share the same dimensional formula [M⁰ L T⁻¹].
Dimensional formulae are essential tools for deriving relationships, checking equation consistency, and converting units between different systems.

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Question 4

Long Form

Explain the principle of homogeneity of dimensions. Using this principle, verify whether the kinematic equation x = x₀ + v₀t + ½at² is dimensionally consistent.

The principle of homogeneity states that every term in a physically correct equation must have the same dimensions; only dimensionally alike quantities can be added or subtracted.
For the equation x = x₀ + v₀t + (1/2)at², the dimension of x (displacement) is [L].
Dimension of x₀ is [L]; dimension of v₀t = [L T⁻¹][T] = [L]; dimension of (1/2)at² = [L T⁻²][T²] = [L].
Since all four terms — x, x₀, v₀t, and (1/2)at² — have the dimension [L], the equation passes the homogeneity test and is dimensionally consistent.
Passing this test proves the equation cannot be wrong on dimensional grounds, but it does not guarantee the equation is exactly correct because dimensionless coefficients are undetectable.
If any term had dimensions different from [L], the equation would be proved wrong immediately; this makes dimensional checking a quick and powerful diagnostic tool.

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Question 5

Long Form

Using dimensional analysis, derive the expression for the time period of a simple pendulum, assuming it depends on the length of the pendulum, the mass of the bob, and acceleration due to gravity.

Assume T = k lˣ gʸ mᶻ, where k is a dimensionless constant and x, y, z are exponents to be found by dimensional analysis.
Writing dimensions on both sides: [M⁰ L⁰ T¹] = [L]ˣ [L T⁻²]ʸ [M]ᶻ = L^(x+y) T^(−2y) M^z.
Equating exponents: for M: z = 0; for T: −2y = 1, giving y = −1/2; for L: x + y = 0, giving x = 1/2.
Substituting: T = k l^(1/2) g^(−1/2) = k√(l/g).
Dimensional analysis cannot determine k; experimental and theoretical work shows k = 2π, giving T = 2π√(l/g).
This result shows the period is independent of the mass of the bob (z = 0) and increases with pendulum length, both of which are confirmed experimentally.

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Question 6

Long Form

Discuss the rules for rounding off uncertain digits in a measured result. Illustrate with examples how rounding affects the final answer in multi-step calculations.

When the digit to be dropped is greater than 5, the preceding digit is raised by 1; when it is less than 5, the preceding digit is left unchanged.
When the digit to be dropped is exactly 5, the convention is: if the preceding digit is even, drop the 5 and leave it unchanged; if the preceding digit is odd, raise it by 1.
Example: 2.746 rounded to three significant figures is 2.75 (digit 6 > 5); 1.743 rounded to three significant figures is 1.74 (digit 3 < 5).
Example: 2.745 rounded to three significant figures is 2.74 (preceding digit 4 is even); 2.735 rounded to three significant figures is 2.74 (preceding digit 3 is odd, raised to 4).
In multi-step calculations, retaining one extra digit in intermediate steps prevents error accumulation; for example, writing 1/9.58 = 0.1044 before rounding gives a reciprocal of 9.58, but rounding prematurely to 0.104 gives 9.62.
The speed of light 2.99792458 × 10⁸ m/s is often rounded to 3 × 10⁸ m/s for calculations, but exact values must be used when precision matters.

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Question 7

Long Form

Compare the three applications of dimensional analysis: checking dimensional consistency, deducing relations, and converting units between systems. State the limitations of each.

Checking consistency uses the principle of homogeneity: all terms in an equation must have identical dimensions. It immediately identifies wrong equations but cannot confirm correctness due to unknown dimensionless constants.
Deducing relations assumes the physical quantity is a product of powers of relevant base quantities (up to three); exponents are found by equating dimensions on both sides of the assumed equation.
Unit conversion uses dimensional formulae to establish the ratio between units in different systems; for example, converting from CGS to SI by substituting base unit relationships.
Limitation of consistency check: a dimensionally correct equation such as K = (3/16)mv² may still be physically wrong; the method cannot detect errors hidden in dimensionless coefficients.
Limitation of deducing relations: the method fails when more than three independent base quantities are involved and cannot be applied when the relationship is not of the simple product-power type.
Common limitation: the method gives no information about dimensionless quantities, pure numbers, or functions (e.g., sin, log, exp), and it cannot distinguish between quantities sharing the same dimensional formula.

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Long Answer

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Hard

Question 8

Long Form

Analyse the concept of measurement errors as reflected in significant figures. How does the combination of errors rule apply to products, quotients, and sums of measured quantities?

Every measurement has a limited precision; significant figures encode this precision — the last significant digit carries an uncertainty of approximately ±1 in its place.
The relative error of a measurement depends on both the number of significant figures and the value itself; 1.02 g has ~±1% relative error while 9.89 g measured to the same precision has only ~±0.1%.
For a product or quotient, the relative errors of the factors add; the final result is quoted to as many significant figures as the factor with the fewest, since this limits the overall precision.
For a sum or difference, absolute uncertainties add; hence the result is rounded to the same number of decimal places as the term with the fewest decimal places.
Subtraction can severely reduce significant figures; for example, 12.9 g − 7.06 g = 5.8 g (only two sig. figs.) despite both inputs having three significant figures.
To minimise cumulative rounding error in multi-step calculations, one extra digit beyond the significant figures should be retained at each intermediate stage and the final result rounded only at the end.

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Long Answer

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Question 9

Long Form

Explain the seven SI base units with their names, symbols, and the physical constants used to define them as per the 2018 revision.

The metre (m) is defined via the fixed speed of light c = 299792458 m s⁻¹; the kilogram (kg) is defined via the fixed Planck constant h = 6.62607015 × 10⁻³⁴ J s.
The second (s) is defined via the caesium-133 hyperfine frequency Δνcs = 9192631770 Hz; the ampere (A) is defined via the elementary charge e = 1.602176634 × 10⁻¹⁹ C.
The kelvin (K) is defined via the Boltzmann constant k = 1.380649 × 10⁻²³ J K⁻¹; the mole (mol) is defined such that one mole contains exactly 6.02214076 × 10²³ elementary entities (Avogadro number).
The candela (cd) is defined via the luminous efficacy of 540 × 10¹² Hz monochromatic radiation Kcd = 683 lm W⁻¹.
The 2018 revision anchored all seven base units to fixed numerical values of universal physical constants, eliminating the last physical artefact (the International Prototype of the Kilogram).
This approach ensures that any laboratory equipped with appropriate technology can realise the base units independently, guaranteeing long-term stability and global reproducibility.

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Long Answer

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Hard

Question 10

Long Form

Using dimensional analysis, evaluate which of the following formulae for kinetic energy K can be ruled out: (a) K = m²v³, (b) K = (1/2)mv², (c) K = ma, (d) K = (3/16)mv², (e) K = (1/2)mv² + ma. Justify each decision.

The SI unit of kinetic energy is J = kg m² s⁻², so the correct formula must have dimensions [M L² T⁻²].
Formula (a): dimensions of m²v³ = [M²][L³ T⁻³] = [M² L³ T⁻³] ≠ [M L² T⁻²]; ruled out.
Formula (b): dimensions of (1/2)mv² = [M][L² T⁻²] = [M L² T⁻²]; dimensionally correct, cannot be ruled out.
Formula (c): dimensions of ma = [M][L T⁻²] = [M L T⁻²] ≠ [M L² T⁻²]; ruled out.
Formula (d): dimensions of (3/16)mv² = [M L² T⁻²]; dimensionally correct, cannot be ruled out by dimensional analysis alone.
Formula (e): (1/2)mv² has dimensions [M L² T⁻²] while ma has dimensions [M L T⁻²]; adding quantities of different dimensions violates the principle of homogeneity, so (e) is ruled out. Only the actual definition of kinetic energy confirms (b), not (d).

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Chapter 1: Units and Measurement | Long Questions

Describe the SI system of units. Why was it developed, and what are its key features that make it superior to earlier systems?

Explain the concept of significant figures. Discuss the rules for determining significant figures and the arithmetic rules for operations involving them.

Describe the concept of dimensions of physical quantities. Derive the dimensional formulae for velocity, force, and mass density, and write their dimensional equations.

Explain the principle of homogeneity of dimensions. Using this principle, verify whether the kinematic equation x = x₀ + v₀t + ½at² is dimensionally consistent.

Using dimensional analysis, derive the expression for the time period of a simple pendulum, assuming it depends on the length of the pendulum, the mass of the bob, and acceleration due to gravity.

Discuss the rules for rounding off uncertain digits in a measured result. Illustrate with examples how rounding affects the final answer in multi-step calculations.

Compare the three applications of dimensional analysis: checking dimensional consistency, deducing relations, and converting units between systems. State the limitations of each.

Analyse the concept of measurement errors as reflected in significant figures. How does the combination of errors rule apply to products, quotients, and sums of measured quantities?

Explain the seven SI base units with their names, symbols, and the physical constants used to define them as per the 2018 revision.

Using dimensional analysis, evaluate which of the following formulae for kinetic energy K can be ruled out: (a) K = m²v³, (b) K = (1/2)mv², (c) K = ma, (d) K = (3/16)mv², (e) K = (1/2)mv² + ma. Justify each decision.

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