Units and Measurements - Class 11 Physics

Units & Measurements

Overview: Measurement is the foundation of Physics. This chapter establishes the international standards for measurement (SI units) and methods to ensure accuracy and precision (Dimensional Analysis, Significant Figures).

1. Need for Measurement

Physics is a quantitative science. To understand natural phenomena, we must measure physical quantities numerically. Measurement involves comparing a physical quantity with a standard reference called a Unit.

Q = n × u

Where Q is the quantity, n is the numerical value, and u is the unit. (e.g., Length = 5 meters).

2. Systems of Units

Historically, different systems were used:

CGS: Centimetre, Gram, Second
FPS: Foot, Pound, Second
MKS: Metre, Kilogram, Second
SI (Système International): The internationally accepted system today.

Fundamental & Derived Units (SI)

Fundamental Units: Base units that are independent of other units.

Quantity	Unit	Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Electric Current	Ampere	A
Thermodynamic Temp	Kelvin	K
Amount of Substance	Mole	mol
Luminous Intensity	Candela	cd

Derived Units: Units expressed in terms of fundamental units (e.g., Velocity = m/s, Force = kg·m/s² = Newton).

3. Significant Figures

The digits in a measured quantity that are known reliably plus the first uncertain digit.

Rules for counting Sig Figs:

All non-zero digits are significant. (1234 → 4)
Zeros between non-zero digits are significant. (1002 → 4)
Leading zeros are never significant. (0.005 → 1)
Trailing zeros in a number without a decimal point are not significant. (1200 → 2)
Trailing zeros in a number with a decimal point are significant. (2.500 → 4)

Rounding Off Rules

If digit to drop < 5, leave preceding digit unchanged.
If digit to drop > 5, increase preceding digit by 1.
If digit to drop = 5 followed by non-zeros, increase preceding by 1.
If digit to drop = 5 followed by zeros:
- If preceding digit is even, leave unchanged.
- If preceding digit is odd, increase by 1.

4. Dimensions of Physical Quantities

The powers to which the fundamental units of mass [M], length [L], and time [T] must be raised to represent a physical quantity.

Examples:

Velocity: [LT^-1]
Force: [MLT^-2]
Energy: [ML²T^-2]

5. Dimensional Analysis & Applications

Principle of Homogeneity: Only physical quantities of the same dimensions can be added, subtracted, or equated.

Applications:

To check the correctness of an equation:
LHS dimensions must equal RHS dimensions.
e.g., s = ut + ½at² → [L] = [LT^-1][T] + [LT^-2][T²] → [L] = [L] + [L]. (Correct)
To convert units from one system to another:
n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c
To derive relations between physical quantities:
If a quantity depends on factors, we can assume powers (a, b, c) and compare dimensions to solve for them. (e.g., Time period of pendulum T ∝ l^a g^b).

Limitations

Does not give information about dimensionless constants (like ½, π).
Cannot derive relations containing trigonometric, exponential, or log functions.
Fails if a quantity depends on more than 3 fundamental factors (in Mechanics).

Numericals - Units and Measurements

Numericals

Sig Figs

Q1. State the number of significant figures in the following: (a) 0.007 m² (b) 2.64 × 10²⁴ kg (c) 0.2370 g

(a) 0.007 m²

Leading zeros are not significant.

Sig Figs = 1

(b) 2.64 × 10²⁴ kg

Power of 10 is irrelevant.

Sig Figs = 3

Trailing zero after decimal is significant.

Sig Figs = 4

Error Analysis

Q2. The resistance R = V/I. where V = (100 ± 5)V and I = (10 ± 0.2)A. Find the percentage error in R.

Percentage error in V:

%V = (5/100) × 100 = 5%

Percentage error in I:

%I = (0.2/10) × 100 = 2%

Total error in division adds up:

%R = %V + %I

%R = 5% + 2%

%R = 7%

Dimensional Consistency

Q3. Check the correctness of the relation v² - u² = 2as where symbols have their usual meanings.

Dimensions of LHS:

[v] = [LT^-1] → [v²] = [L²T^-2]

[u] = [LT^-1] → [u²] = [L²T^-2]

Dimensions of RHS:

[a] = [LT^-2], [s] = [L]

[2as] = [LT^-2][L] = [L²T^-2]

(2 is dimensionless)

LHS = RHS = [L²T^-2]

Hence, the equation is dimensionally correct.

Unit Conversion

Q4. Convert 1 Joule into ergs using dimensional analysis.

Joule is SI, Erg is CGS unit of Energy.

Dim of Energy = [ML²T^-2]

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

n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c

n₂ = 1 [kg/g]¹ [m/cm]² [s/s]^-2

n₂ = 1 [1000g/g] [100cm/cm]² [1]

n₂ = 1 × 1000 × (100)²

n₂ = 10³ × 10⁴

n₂ = 10⁷

1 Joule = 10⁷ ergs

Significant Figures

Q5. Keep the correct number of significant figures: 5.74 g of a substance occupies 1.2 cm³. Find density.

Density = Mass / Volume

D = 5.74 / 1.2

D = 4.78333... g/cm³

Least sig figs in input is 2 (from 1.2).

Result must be rounded to 2 sig figs.

D = 4.8 g/cm³

Dimensional Derivation

Q6. The time period T of a simple pendulum depends on length l and gravity g. Derive the formula. (Take constant k=2π)

T ∝ l^a g^b

Dimensions:

[T] = [L]^a [LT^-2]^b

[M⁰L⁰T¹] = [L^a+b T^-2b]

Comparing powers:

-2b = 1 → b = -1/2

a + b = 0 → a = -b = 1/2

T = k l^1/2 g^-1/2

T = 2π √(l/g)

Volume Calculation

Q7. The side of a cube is 7.203 m. Calculate its volume with appropriate sig figs.

Side a = 7.203 m (4 Sig Figs)

V = a³

V = 7.203 × 7.203 × 7.203

V = 373.714754... m³

Round to 4 sig figs:

V = 373.7 m³

Powers of 10

Q8. Add 3.8 × 10^-6 and 4.2 × 10^-5 with regards to significant figures.

Convert to same power of 10 (usually larger).

A = 0.38 × 10^-5

B = 4.2 × 10^-5

Sum = (0.38 + 4.2) × 10^-5

Sum = 4.58 × 10^-5

Rule for addition: Least decimal places. 4.2 has 1 decimal place. 0.38 has 2. Result needs 1.

Round 4.58 → 4.6

Sum = 4.6 × 10^-5

Derived Dimensions

Q9. Find the dimensions of Planck's Constant (h) from E = hν (where ν is frequency).

h = E / ν

Dim of Energy [E] = [ML²T^-2]

Dim of Frequency [ν] = [T^-1]

[h] = [ML²T^-2] / [T^-1]

[h] = [ML²T^-1]

Precision vs Accuracy

Q10. True value is 3.678. Measurements are 3.5 and 3.38. Compare precision and accuracy.

M1 = 3.5 (Resolution 0.1)

M2 = 3.38 (Resolution 0.01)

Precision:

M2 is more precise (more decimal places).

Accuracy:

Error1 = |3.678 - 3.5| = 0.178

Error2 = |3.678 - 3.38| = 0.298

M1 is closer to true value, hence more accurate.

Formulas & Facts - Units and Measurements

Formulas & Dimensions

Quantity	Formula	Dimensions
Area	l × b	[L²]
Volume	l × b × h	[L³]
Density	Mass / Volume	[ML^-3]
Velocity	Displacement / Time	[LT^-1]
Acceleration	Velocity / Time	[LT^-2]
Force	Mass × Acc.	[MLT^-2]
Work / Energy	Force × Dist	[ML²T^-2]
Power	Work / Time	[ML²T^-3]
Pressure	Force / Area	[ML^-1T^-2]
Frequency	1 / Time Period	[T^-1]

50 NEET Facts

Important points for Units and Measurements.

1. Parsec Parsec is the largest practical unit of distance used in astronomy. It is the distance at which an arc of length 1 astronomical unit subtends an angle of one second of arc. 1 Parsec = 3.08 × 10¹⁶ m.

2. Light Year Light year is a unit of distance, not time. It is the distance traveled by light in vacuum in one year. 1 ly = 9.46 × 10¹⁵ m.

3. Astronomical Unit (AU) AU is the average distance between the Earth and the Sun. 1 AU = 1.496 × 10¹¹ m.

4. Dimensionless Quantities Quantities like angle, solid angle, relative density, strain, refractive index, and trigonometric functions have no dimensions. They are pure numbers.

5. Same Dimensions Pair 1 Work, Energy, Torque, and Heat have the same dimensional formula: [ML²T^-2].

6. Same Dimensions Pair 2 Impulse and Momentum have the same dimensional formula: [MLT^-1].

7. Same Dimensions Pair 3 Pressure, Stress, and Modulus of Elasticity (Young's, Bulk, Shear) have the same dimensions: [ML^-1T^-2].

8. Same Dimensions Pair 4 Angular Velocity, Frequency, and Velocity Gradient have dimensions [T^-1].

9. Principle of Homogeneity We can only add or subtract quantities if they have the same dimensions. This is the basis for checking the consistency of physical equations.

10. Significant Figures Concept Greater the number of significant figures in a measurement, smaller is the percentage error and higher is the precision of the measurement.

11. Unit of Time The Second is defined based on the vibrations of Cesium-133 atom. One second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state.

12. Planck's Constant Dimensions Planck's constant (h) is [ML²T^-1]. This is the same as Angular Momentum.

13. Gravitational Constant (G) The dimensions for Universal Gravitational Constant are [M^-1L³T^-2].

14. Absolute Errors Absolute error is the magnitude of the difference between the true value and the measured value. It always has the same unit as the quantity itself.

15. Relative Error Relative error or fractional error is the ratio of mean absolute error to the mean value of the quantity measured. It is unitless.

16. Steradian Steradian (sr) is the SI unit of solid angle. It is a supplementary unit. It is dimensionless.

17. Vernier Caliper LC Least count of a standard Vernier Caliper is typically 0.01 cm or 0.1 mm. LC = 1 MSD - 1 VSD.

18. Screw Gauge LC Least count of a standard Screw Gauge is typically 0.001 cm or 0.01 mm. LC = Pitch / Number of circular scale divisions.

19. Accuracy vs Precision Accuracy refers to how close a measurement is to the true value. Precision refers to the resolution or limit of the quantity measured (closeness of multiple measurements).

20. Systematic Errors Errors that tend to be in one direction (either positive or negative). Causes include instrumental errors, imperfection in technique, etc. Can be minimized by correction factors.

21. Random Errors Errors which occur irregularly and are random with respect to sign and size. Can be reduced by taking the arithmetic mean of a large number of observations.

22. Dimensional Limitations Dimensional analysis cannot be used to derive formulae containing trigonometric, exponential, or logarithmic functions as they are dimensionless.

23. Coeff. of Viscosity Dimensions of Coefficient of Viscosity (η) are [ML^-1T^-1].

24. Solar Day vs Sidereal Day Solar day is the time interval between two successive noons. Sidereal day is the time interval between two successive transits of a distant star. Sidereal day is shorter by about 4 mins.

25. Shake A 'Shake' is an informal unit of time equal to 10^-8 seconds, sometimes used in nuclear physics.

26. Chandrasekhar Limit CSL is the largest unit of mass. 1 CSL = 1.4 times the mass of the Sun.

27. Barn Barn is a unit of area used in nuclear physics to measure nuclear cross-sections. 1 Barn = 10^-28 m².

28. Error Propagation (Sum/Diff) When two quantities are added or subtracted, the absolute error in the final result is the sum of the individual absolute errors. ΔZ = ΔA + ΔB.

29. Error Propagation (Product/Div) When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers. ΔZ/Z = ΔA/A + ΔB/B.

30. Error with Powers If Z = Aⁿ, then relative error ΔZ/Z = n(ΔA/A). Power factor multiplies the relative error.

31. 1 AMU 1 Atomic Mass Unit (amu or u) = 1.66 × 10^-27 kg. Defined as 1/12th of mass of C-12 atom.

32. Least Count Error Associated with the resolution of the instrument. It is a random error but can be reduced by using higher precision instruments.

33. Fundamental Quantities Count There are exactly 7 fundamental quantities in SI system. All others are derived.

34. Supplementary Units Radian (for plane angle) and Steradian (for solid angle). They have units but no dimensions.

35. Fermi Fermi is a unit of length used to measure nuclear distances. 1 Fermi (fm) = 10^-15 m.

36. Micron Micron (μm) is 10^-6 m. Common in biology and microscopy.

37. Surface Tension Dim Surface Tension = Force / Length. Dimensions: [MT^-2].

38. Thermal Conductivity Dimensions of k are [MLT^-3K^-1].

39. Boltzmann Constant Dimensions of k_B are [ML²T^-2K^-1]. Same as Entropy.

40. Angstrom Unit of length for atomic size and wavelength. 1 Å = 10^-10 m.

41. Volt Dimensions Potential = Work / Charge. Dimensions: [ML²T^-3A^-1].

42. Resistance Dimensions R = V/I. Dimensions: [ML²T^-3A^-2].

43. Capacitance Dimensions C = Q/V. Dimensions: [M^-1L^-2T⁴A²].

44. Inductance Dimensions L dimensions: [ML²T^-2A^-2].

45. Order of Magnitude The power of 10 closest to the magnitude of value. e.g for 500 (5 × 10²), order is 10³ because 5 ≥ √10.

46. Specific Heat Capacity Dimensions: [L²T^-2K^-1]. Independent of Mass dimension.

47. Latent Heat Dimensions: [L²T^-2]. Energy per unit mass.

48. Zero Error A type of systematic error where instrument reads a value when it should read zero.

49. Mass vs Weight Mass [M] is fundamental. Weight is Force [MLT^-2].

50. Constants with Dimensions Gravitational constant (G), Speed of light (c), Planck's constant (h) are dimensional constants.

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