Units and Measurements

What to remember

• Every physical quantity is a number times a unit. The SI system has seven base quantities: length (m), mass (kg), time (s), electric current (A), temperature (K), amount of substance (mol) and luminous intensity (cd).
• Significant figures show the precision of a measurement. In multiplication and division, the result has as many significant figures as the least precise factor. In addition and subtraction, the result has as many decimal places as the least precise term.
• Errors: in a sum or difference the absolute errors add. In a product or quotient the relative (fractional) errors add. For a power x^n, the relative error is n times the relative error of x.
• Dimensional analysis checks equations, converts units and derives relations, but it cannot give numerical constants and it fails for sums of terms, trigonometric, logarithmic and exponential functions of dimensional arguments.

Need for measurement and units

Physics is a science of measurement. To measure a quantity, we compare it with a standard of the same kind. This standard is called a unit. The result of a measurement has two parts: a number and a unit. For example, a length of 5 m means 5 times the unit metre.

Some quantities are chosen as fundamental (base) quantities. All other quantities are expressed through them and are called derived quantities. The units of the base quantities are base units. The units of derived quantities are derived units. A complete set of base and derived units is a system of units.

The system used worldwide is the International System of Units (SI). It has seven base units and two supplementary units.

A plane angle in radians is the arc length divided by the radius: θ = s/r. A solid angle in steradians is the area on a sphere divided by the square of the radius: Ω = A/r². Both are ratios of like quantities, so they have no dimensions.

Some derived units have special names: newton (N = kg m s⁻²) for force, joule (J = N m) for energy, watt (W = J s⁻¹) for power, pascal (Pa = N m⁻²) for pressure, hertz (Hz = s⁻¹) for frequency, coulomb (C = A s) for charge.

Common prefixes: kilo (10³), mega (10⁶), giga (10⁹), milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹), pico (10⁻¹²).

Other useful units: 1 light year ≈ 9.46 × 10¹⁵ m, 1 astronomical unit ≈ 1.5 × 10¹¹ m, 1 angstrom = 10⁻¹⁰ m, 1 fermi = 10⁻¹⁵ m, 1 atomic mass unit ≈ 1.66 × 10⁻²⁷ kg.

Significant figures

Significant figures are the reliable digits in a measured value plus the first uncertain digit. They show the precision of the measurement.

Rules for counting:

• 1. All non-zero digits are significant (2.34 has three).
• 2. Zeros between non-zero digits are significant (3.007 has four).
• 3. Zeros to the left of the first non-zero digit are not significant (0.0052 has two).
• 4. Trailing zeros after a decimal point are significant (4.50 has three; 0.0500 has three).
• 5. Trailing zeros in a number without a decimal point are not clearly significant. Scientific notation removes the doubt: 1.20 × 10³ has three, 1.2 × 10³ has two.
• 6. Changing the unit does not change the number of significant figures: 2.30 m = 230 cm = 2.30 × 10⁻³ km.
• 7. Exact numbers (counts, such as 5 oscillations, or defined factors such as 2 in 2πr) have unlimited significant figures.

Rules for arithmetic:

• Multiplication and division: the result keeps as many significant figures as the factor with the fewest significant figures. Example: 4.2 × 3.14 = 13.188, written as 13 (two significant figures).
• Addition and subtraction: the result keeps as many decimal places as the term with the fewest decimal places. Example: 12.3 + 0.456 = 12.756, written as 12.8.

Rules for rounding off:

• If the digit to be dropped is less than 5, the preceding digit is unchanged (2.43 → 2.4).
• If it is more than 5, the preceding digit is raised by one (2.47 → 2.5).
• If it is exactly 5 with nothing after it, the preceding digit is left unchanged when it is even and raised by one when it is odd (2.45 → 2.4; 2.35 → 2.4).

Errors in measurement

The difference between the measured value and the true value is the error. Error is not a mistake. Even a careful measurement has some error.

• Systematic errors have a definite direction and size. Examples: zero error of an instrument, wrong calibration, personal bias.
• Random errors vary in size and sign, and arise from unknown causes. Repeating the measurement and taking the mean reduces them.
• Least count error: the least count is the smallest value an instrument can measure. It sets the limit of the instrument's precision.

Definitions, for n readings a₁, a₂, ... aₙ:

• Mean value: a_mean = (a₁ + a₂ + ... + aₙ)/n. It is taken as the best estimate of the true value.
• Absolute error of one reading: Δaᵢ = aᵢ − a_mean (its magnitude is used).
• Mean absolute error: Δa_mean = (|Δa₁| + |Δa₂| + ... + |Δaₙ|)/n.
• Relative error (fractional error) = Δa_mean / a_mean. It has no unit.
• Percentage error = (Δa_mean / a_mean) × 100 %.

Combination of errors (worst case):

Note that errors always add, even in a difference. The power n in the last row is used as a magnitude.

Worked example 1 (sum and difference). Two lengths are L₁ = 10.5 ± 0.1 cm and L₂ = 26.9 ± 0.1 cm. The sum is 37.4 cm with error 0.1 + 0.1 = 0.2 cm. The difference is 16.4 cm with error 0.2 cm.

Worked example 2 (powers). The radius of a sphere is measured as 5.0 cm with 2 % error. The volume V = (4/3)πr³ has a relative error 3 × 2 % = 6 %. The surface area 4πr² has a relative error 2 × 2 % = 4 %.

Worked example 3 (product and quotient). Density ρ = m/V is found from m = 20.0 g with 1 % error and V = 4.0 cm³ with 3 % error. The density is 5.0 g cm⁻³ with a percentage error 1 + 3 = 4 %, that is ± 0.2 g cm⁻³.

Worked example 4 (pendulum). For T = 2π√(L/g), g = 4π²L/T². Percentage error in g = (error in L) % + 2 × (error in T) %. If L has 1 % error and T has 2 % error, g has 1 + 4 = 5 % error.

Least count of instruments

The least count is the smallest reading an instrument can give directly.

Vernier callipers: if N vernier divisions equal (N − 1) main scale divisions, then least count = 1 MSD / N. With 1 MSD = 1 mm and N = 10, least count = 0.1 mm.

Screw gauge: pitch is the distance moved by the screw in one full turn. If the pitch is 1 mm and the circular scale has 100 divisions, least count = 1 mm / 100 = 0.01 mm.

Reading = main scale reading + (coinciding division × least count) − zero error. If the instrument shows a zero error, it is subtracted with its sign. Using a thinner measuring object or a finer instrument gives a smaller relative error.

Dimensions of physical quantities

The dimensions of a physical quantity are the powers to which the base quantities are raised to represent it. The base quantities are written in square brackets: [M] mass, [L] length, [T] time, [A] current, [K] temperature, [mol] amount of substance, [cd] luminous intensity.

Quantities with the same dimensions need not be the same kind of quantity. Work and torque both have [M L² T⁻²]. Stress and pressure both have [M L⁻¹ T⁻²]. Frequency and angular velocity both have [T⁻¹].

Dimensional equation: an equation that equates a physical quantity to its dimensional formula, for example [F] = [M L T⁻²].

Dimensional analysis and its uses

Principle of homogeneity. In a correct physical equation, every term has the same dimensions. Only quantities of the same dimensions can be added, subtracted or equated.

Use 1: Checking an equation. Check s = ut + ½at². Dimensions: [s] = [L]; [ut] = [L T⁻¹][T] = [L]; [½at²] = [L T⁻²][T²] = [L]. All terms match, so the equation is dimensionally correct.

Use 2: Converting units. A quantity is n₁u₁ = n₂u₂. Example: convert 1 N into dyne. 1 N = 1 kg m s⁻². In CGS, 1 kg = 10³ g and 1 m = 10² cm. So 1 N = 10³ × 10² g cm s⁻² = 10⁵ dyne. Another example: 1 J = 10⁷ erg.

Use 3: Deriving a relation. Period T of a simple pendulum may depend on its length L, mass m and acceleration due to gravity g. Write T = k mᵃ Lᵇ gᶜ. Dimensions: [T] = [M]ᵃ [L]ᵇ [L T⁻²]ᶜ. Equate powers: M: a = 0. L: b + c = 0. T: −2c = 1, so c = −½ and b = ½. Hence T = k √(L/g). The mass does not appear. The constant k = 2π is found only by experiment or theory.

Limitations of dimensional analysis.

• 1. It cannot give the value of dimensionless constants such as 2π or ½.
• 2. It cannot be used when a quantity depends on more than three other quantities in a way that cannot be separated as a product of powers (more unknowns than equations).
• 3. It fails for relations containing sums of terms, for example s = ut + ½at² cannot be derived by it. It also cannot decide between different forms with the same dimensions.
• 4. It cannot be applied to trigonometric, logarithmic and exponential functions, whose arguments must be dimensionless.
• 5. A dimensionally correct equation need not be physically correct. For example, K = mv² is dimensionally correct but the correct kinetic energy is ½mv².

Formula summary

NEET traps

• 1. Absolute errors add in a difference too. They are never subtracted.
• 2. Relative error has no unit; absolute error has the unit of the quantity.
• 3. Zero error is subtracted with its sign: a positive zero error means the reading is too large, so subtract it.
• 4. Trailing zeros without a decimal point (such as 200) are not clearly significant; 200. or 2.00 × 10² has three.
• 5. Leading zeros (0.0042) are never significant.
• 6. Torque and work have the same dimensions but are different quantities; the same is true for frequency and angular velocity.
• 7. Strain, angle, refractive index and relative density are dimensionless but may or may not have a unit (radian is a unit for angle).
• 8. A dimensionally correct equation can still be wrong: K = mv² passes the dimensional test.
• 9. Dimensional analysis gives no constants, and cannot derive a relation that is a sum of terms.
• 10. The kilogram is the SI base unit of mass, not the gram. Prefixes are applied to the gram when naming, but the base unit is kg.

One-liners

• 1. SI has seven base quantities and two supplementary quantities (plane angle, solid angle).
• 2. The dimensional formula of force is [M L T⁻²]; of energy, [M L² T⁻²].
• 3. Planck's constant and angular momentum both have dimensions [M L² T⁻¹].
• 4. The dimensions of G are [M⁻¹ L³ T⁻²].
• 5. Pressure and Young's modulus have dimensions [M L⁻¹ T⁻²].
• 6. 1 N = 10⁵ dyne and 1 J = 10⁷ erg.
• 7. 1 light year ≈ 9.46 × 10¹⁵ m.
• 8. 1 fermi = 10⁻¹⁵ m and 1 angstrom = 10⁻¹⁰ m.
• 9. The number of significant figures does not change when the unit is changed.
• 10. For Z = A B, the percentage error in Z is the sum of the percentage errors in A and B.
• 11. For Z = A³, the percentage error in Z is three times that in A.
• 12. The least count of a vernier callipers = 1 MSD − 1 VSD.
• 13. The least count of a screw gauge = pitch / number of circular scale divisions.
• 14. The period of a simple pendulum does not depend on the mass of the bob, as dimensional analysis shows.
• 15. The arguments of sin, log and exponential functions must be dimensionless.

Practice questions

1. What is the SI unit of luminous intensity?

   1. lumen
   2. watt
   3. lux
   4. candela
   Answer

   D. candela

   Luminous intensity is a base quantity; its SI base unit is the candela (cd).

2. The dimensional formula of pressure is

   1. [M L⁻² T⁻²]
   2. [M L T⁻²]
   3. [M L⁻¹ T⁻²]
   4. [M L² T⁻²]
   Answer

   C. [M L⁻¹ T⁻²]

   Pressure = force/area = [M L T⁻²]/[L²] = [M L⁻¹ T⁻²].

3. How many significant figures does the number 0.04050 have?

   1. 5
   2. 4
   3. 3
   4. 2
   Answer

   B. 4

   Leading zeros are not significant; the digits 4, 0, 5, 0 are significant (the trailing zero after the decimal point counts). So 4.

4. The sum 12.11 + 18.0 + 1.012, written with the correct number of significant figures, is

   1. 31
   2. 31.12
   3. 31.122
   4. 31.1
   Answer

   D. 31.1

   Sum = 31.122. The least precise term (18.0) has one decimal place, so the result is 31.1.

5. A quantity is given by Z = A²B³/C. The percentage errors in A, B and C are 1 %, 2 % and 3 % respectively. The percentage error in Z is

   1. 6 %
   2. 5 %
   3. 11 %
   4. 8 %
   Answer

   C. 11 %

   Relative errors add with powers as multipliers: 2(1) + 3(2) + 1(3) = 2 + 6 + 3 = 11 %.

6. The dimensional formula of Planck's constant is

   1. [M L³ T⁻¹]
   2. [M L² T⁻¹]
   3. [M L² T⁻²]
   4. [M L T⁻¹]
   Answer

   B. [M L² T⁻¹]

   h = E/ν, so [h] = [M L² T⁻²]/[T⁻¹] = [M L² T⁻¹].

7. The dimensional formula of the universal gravitational constant G is

   1. [M⁻¹ L³ T⁻²]
   2. [M⁻² L³ T⁻²]
   3. [M⁻¹ L² T⁻²]
   4. [M L³ T⁻²]
   Answer

   A. [M⁻¹ L³ T⁻²]

   From F = Gm₁m₂/r², G = F r²/(m₁m₂) = [M L T⁻²][L²]/[M²] = [M⁻¹ L³ T⁻²].

8. The number 4.65 is rounded off to two significant figures. The result is

   1. 4.5
   2. 4.7
   3. 4.6
   4. 4.8
   Answer

   C. 4.6

   The digit to be dropped is exactly 5 and the preceding digit 6 is even, so it is left unchanged: 4.6.

9. In a vernier callipers, 20 vernier scale divisions coincide with 19 main scale divisions. One main scale division is 1 mm. The least count is

   1. 0.01 mm
   2. 0.05 mm
   3. 0.1 mm
   4. 0.5 mm
   Answer

   B. 0.05 mm

   1 VSD = 19/20 mm. Least count = 1 MSD − 1 VSD = 1 − 0.95 = 0.05 mm.

10. A screw gauge has a pitch of 0.5 mm and 50 divisions on its circular scale. Its least count is

    1. 0.01 mm
    2. 0.5 mm
    3. 0.1 mm
    4. 0.001 mm
    Answer

    A. 0.01 mm

    Least count = pitch / number of divisions = 0.5/50 = 0.01 mm.

11. Which pair of quantities has the same dimensions?

    1. Force and impulse
    2. Pressure and energy
    3. Frequency and velocity
    4. Work and torque
    Answer

    D. Work and torque

    Work and torque both have dimensions [M L² T⁻²]. Impulse is [M L T⁻¹], and the other pairs differ.

12. Which of the following quantities is dimensionless?

    1. Young's modulus
    2. Refractive index
    3. Surface tension
    4. Angular velocity
    Answer

    B. Refractive index

    Refractive index is a ratio of two speeds, so it has no dimensions. The others have dimensions.

13. Dimensional analysis cannot determine

    1. whether an equation is dimensionally consistent
    2. the dimensions of a physical constant
    3. dimensionless numerical constants in a formula
    4. the units of a derived quantity
    Answer

    C. dimensionless numerical constants in a formula

    Constants such as 2π or ½ are dimensionless, so dimensions cannot fix them.

14. One light year is approximately equal to

    1. 9.46 × 10¹⁵ m
    2. 3 × 10⁸ m
    3. 9.46 × 10¹² m
    4. 1.5 × 10¹¹ m
    Answer

    A. 9.46 × 10¹⁵ m

    Light travels 3 × 10⁸ m s⁻¹ for about 3.15 × 10⁷ s in a year, about 9.46 × 10¹⁵ m.

15. The dimensional formula [M L² T⁻³] belongs to

    1. force
    2. energy
    3. momentum
    4. power
    Answer

    D. power

    Power = work/time = [M L² T⁻²]/[T] = [M L² T⁻³].

16. Given below are two statements: Statement I: In a difference of two measured quantities, the absolute errors add. Statement II: In a quotient of two measured quantities, the relative errors add.

    1. Both Statement I and Statement II are correct
    2. Both Statement I and Statement II are incorrect
    3. Statement I is correct but Statement II is incorrect
    4. Statement I is incorrect but Statement II is correct
    Answer

    A. Both Statement I and Statement II are correct

    Both statements are correct: absolute errors add in sums and differences; relative errors add in products and quotients.

17. Given below are two statements: Statement I: The number 0.0300 has three significant figures. Statement II: Leading zeros in a decimal number are significant.

    1. Both Statement I and Statement II are correct
    2. Both Statement I and Statement II are incorrect
    3. Statement I is correct but Statement II is incorrect
    4. Statement I is incorrect but Statement II is correct
    Answer

    C. Statement I is correct but Statement II is incorrect

    0.0300 has digits 3, 0, 0 significant, so three. Leading zeros are not significant, so Statement II is incorrect.

18. Given below are two statements: Statement I: The kilogram, the second and the radian are all SI base units. Statement II: The SI unit of plane angle is the radian.

    1. Both Statement I and Statement II are correct
    2. Both Statement I and Statement II are incorrect
    3. Statement I is correct but Statement II is incorrect
    4. Statement I is incorrect but Statement II is correct
    Answer

    D. Statement I is incorrect but Statement II is correct

    The radian is a supplementary unit, not a base unit, so Statement I is incorrect. Statement II is correct.

19. Given below are two statements: Statement I: Dimensional analysis can give the value 2π in the formula for the period of a simple pendulum. Statement II: The period of a simple pendulum depends on the mass of the bob, as shown by dimensions.

    1. Both Statement I and Statement II are correct
    2. Both Statement I and Statement II are incorrect
    3. Statement I is correct but Statement II is incorrect
    4. Statement I is incorrect but Statement II is correct
    Answer

    B. Both Statement I and Statement II are incorrect

    Dimensional analysis cannot give constants like 2π, and it shows that the mass cancels out. Both statements are incorrect.

20. Given below are two statements: Statement I: A plane angle is measured in radian and a solid angle in steradian. Statement II: Solid angle is the ratio of a length to the radius.

    1. Both Statement I and Statement II are correct
    2. Both Statement I and Statement II are incorrect
    3. Statement I is correct but Statement II is incorrect
    4. Statement I is incorrect but Statement II is correct
    Answer

    C. Statement I is correct but Statement II is incorrect

    Statement I is correct. Solid angle = area/r², not length/r, so Statement II is incorrect.

21. Which of the following statements are correct? 1. The number of significant figures does not change when the unit is changed. 2. Exact counts such as 5 oscillations have unlimited significant figures. 3. In multiplication, the result has as many significant figures as the least precise factor.

    1. 1 and 2 only
    2. 2 and 3 only
    3. 1 and 3 only
    4. 1, 2 and 3
    Answer

    D. 1, 2 and 3

    All three statements are correct rules for significant figures.

22. Which of the following statements are correct? 1. Force and impulse have the same dimensions. 2. Surface tension has dimensions [M T⁻²]. 3. Frequency and angular velocity have the same dimensions.

    1. 1 and 2 only
    2. 2 and 3 only
    3. 1 and 3 only
    4. 1, 2 and 3
    Answer

    B. 2 and 3 only

    Force is [M L T⁻²] and impulse is [M L T⁻¹], so 1 is wrong. Surface tension [M T⁻²] and [T⁻¹] for both frequency and angular velocity are correct.

23. Given below are two statements: Statement I: If L has 2 % error and T has 1 % error, the value of g found from T = 2π√(L/g) has 3 % error. Statement II: In a quotient, the relative errors of the quantities add.

    1. Both Statement I and Statement II are correct
    2. Both Statement I and Statement II are incorrect
    3. Statement I is correct but Statement II is incorrect
    4. Statement I is incorrect but Statement II is correct
    Answer

    D. Statement I is incorrect but Statement II is correct

    g = 4π²L/T², so the error is 2 + 2(1) = 4 %, not 3 %. Statement I is incorrect; Statement II is correct.

24. Given below are two statements: Statement I: 2.45 rounded to two significant figures is 2.4. Statement II: 2.55 rounded to two significant figures is 2.6.

    1. Both Statement I and Statement II are correct
    2. Both Statement I and Statement II are incorrect
    3. Statement I is correct but Statement II is incorrect
    4. Statement I is incorrect but Statement II is correct
    Answer

    A. Both Statement I and Statement II are correct

    With a dropped 5, the preceding even digit stays (2.4); the preceding odd digit goes up (2.55 → 2.6). Both correct.

25. Given below are two statements: Statement I: The SI base unit of mass is the gram. Statement II: The SI unit of thermodynamic temperature is the kelvin.

    1. Both Statement I and Statement II are correct
    2. Both Statement I and Statement II are incorrect
    3. Statement I is correct but Statement II is incorrect
    4. Statement I is incorrect but Statement II is correct
    Answer

    D. Statement I is incorrect but Statement II is correct

    The SI base unit of mass is the kilogram, so Statement I is incorrect; Statement II is correct.

26. Assertion (A): The period of a simple pendulum cannot be completely obtained by dimensional analysis. Reason (R): Dimensional analysis cannot give dimensionless constants such as 2π.

    1. Both A and R are true and R is the correct explanation of A
    2. Both A and R are true but R is not the correct explanation of A
    3. A is true but R is false
    4. A is false but R is true
    Answer

    A. Both A and R are true and R is the correct explanation of A

    The constant 2π is left undetermined, so A is true and R explains it.

27. Assertion (A): The relative error in Z = A/B is the sum of the relative errors in A and B. Reason (R): The relative error in Aⁿ is n times the relative error in A.

    1. Both A and R are true and R is the correct explanation of A
    2. Both A and R are true but R is not the correct explanation of A
    3. A is true but R is false
    4. A is false but R is true
    Answer

    B. Both A and R are true but R is not the correct explanation of A

    Both are true rules, but the power rule does not explain the quotient rule.

28. Assertion (A): The number 0.0050 has two significant figures. Reason (R): Zeros to the right of the last non-zero digit after a decimal point are not significant.

    1. Both A and R are true and R is the correct explanation of A
    2. Both A and R are true but R is not the correct explanation of A
    3. A is true but R is false
    4. A is false but R is true
    Answer

    C. A is true but R is false

    0.0050 has digits 5 and 0 significant. Trailing zeros after the decimal point are significant, so R is false.

29. Assertion (A): A dimensionally correct equation is always physically correct. Reason (R): Dimensional analysis cannot determine numerical factors such as ½.

    1. Both A and R are true and R is the correct explanation of A
    2. Both A and R are true but R is not the correct explanation of A
    3. A is true but R is false
    4. A is false but R is true
    Answer

    D. A is false but R is true

    K = mv² is dimensionally correct but wrong. A is false and R is true.

30. Assertion (A): For measuring the thickness of a thin wire, a screw gauge gives a smaller relative error than a metre scale. Reason (R): The least count of a screw gauge is much smaller than that of a metre scale.

    1. Both A and R are true and R is the correct explanation of A
    2. Both A and R are true but R is not the correct explanation of A
    3. A is true but R is false
    4. A is false but R is true
    Answer

    A. Both A and R are true and R is the correct explanation of A

    Relative error = least count/measured value; a smaller least count gives a smaller relative error.

31. Match List I with List II. List I: a. Force b. Pressure c. Power d. Impulse List II: i. [M L⁻¹ T⁻²] ii. [M L T⁻¹] iii. [M L² T⁻³] iv. [M L T⁻²]

    1. a-iv, b-i, c-ii, d-iii
    2. a-ii, b-i, c-iii, d-iv
    3. a-iv, b-i, c-iii, d-ii
    4. a-i, b-iv, c-iii, d-ii
    Answer

    C. a-iv, b-i, c-iii, d-ii

    Force [M L T⁻²], pressure [M L⁻¹ T⁻²], power [M L² T⁻³], impulse [M L T⁻¹]: a-iv, b-i, c-iii, d-ii.

32. Match List I with List II. List I: a. Metre scale b. Vernier callipers (20 VSD = 19 MSD, 1 MSD = 1 mm) c. Screw gauge (pitch 1 mm, 100 divisions) d. Vernier callipers (10 VSD = 9 MSD, 1 MSD = 1 mm) List II: i. 0.05 mm ii. 0.01 mm iii. 1 mm iv. 0.1 mm

    1. a-iv, b-ii, c-i, d-iii
    2. a-iii, b-i, c-iv, d-ii
    3. a-iii, b-i, c-ii, d-iv
    4. a-i, b-iii, c-ii, d-iv
    Answer

    C. a-iii, b-i, c-ii, d-iv

    Metre scale 1 mm; 1/20 = 0.05 mm; 1/100 = 0.01 mm; 1/10 = 0.1 mm.

33. Match List I with List II. List I: a. Plane angle b. Solid angle c. Luminous intensity d. Amount of substance List II: i. mole ii. radian iii. candela iv. steradian

    1. a-i, b-iv, c-iii, d-ii
    2. a-ii, b-iv, c-iii, d-i
    3. a-ii, b-iv, c-i, d-iii
    4. a-iv, b-ii, c-iii, d-i
    Answer

    B. a-ii, b-iv, c-iii, d-i

    Units: radian, steradian, candela, mole.

34. Match List I with List II. List I: a. 0.0360 b. 3.0060 c. 2.0 × 10³ d. 0.005 List II: i. 1 significant figure ii. 2 significant figures iii. 3 significant figures iv. 5 significant figures

    1. a-iii, b-iv, c-ii, d-i
    2. a-iii, b-iv, c-i, d-ii
    3. a-iii, b-ii, c-iv, d-i
    4. a-iv, b-iii, c-ii, d-i
    Answer

    A. a-iii, b-iv, c-ii, d-i

    0.0360 has 3, 3.0060 has 5, 2.0 × 10³ has 2, 0.005 has 1.

35. Match List I with List II. List I: a. Planck's constant b. Universal gravitational constant c. Surface tension d. Coefficient of viscosity List II: i. [M T⁻²] ii. [M L⁻¹ T⁻¹] iii. [M L² T⁻¹] iv. [M⁻¹ L³ T⁻²]

    1. a-i, b-iv, c-iii, d-ii
    2. a-iv, b-iii, c-i, d-ii
    3. a-iii, b-iv, c-i, d-ii
    4. a-iii, b-iv, c-ii, d-i
    Answer

    C. a-iii, b-iv, c-i, d-ii

    h [M L² T⁻¹]; G [M⁻¹ L³ T⁻²]; surface tension [M T⁻²]; viscosity [M L⁻¹ T⁻¹].

36. Match List I with List II. List I: a. Z = A + B b. Z = AB c. Z = A³ d. Z = A/B² List II: i. ΔZ = ΔA + ΔB ii. ΔZ/Z = ΔA/A + ΔB/B iii. ΔZ/Z = 3 ΔA/A iv. ΔZ/Z = ΔA/A + 2 ΔB/B

    1. a-i, b-ii, c-iii, d-iv
    2. a-iii, b-ii, c-i, d-iv
    3. a-ii, b-i, c-iii, d-iv
    4. a-i, b-ii, c-iv, d-iii
    Answer

    A. a-i, b-ii, c-iii, d-iv

    Sum: absolute errors add; product: relative errors add; cube: 3 times; quotient with square: ΔA/A + 2ΔB/B.

37. The acceleration due to gravity is 9.8 m s⁻². Its value in km h⁻² is

    1. 35.28 km h⁻²
    2. 1.27 × 10⁸ km h⁻²
    3. 2.72 km h⁻²
    4. 1.27 × 10⁵ km h⁻²
    Answer

    D. 1.27 × 10⁵ km h⁻²

    9.8 m s⁻² = 9.8 × (3600)²/1000 km h⁻² = 9.8 × 12960 = 127008 ≈ 1.27 × 10⁵ km h⁻².

38. The side of a cube is measured with an error of 1.5 %. The percentage error in its volume is

    1. 2.25 %
    2. 4.5 %
    3. 3.0 %
    4. 1.5 %
    Answer

    B. 4.5 %

    V = a³, so error = 3 × 1.5 % = 4.5 %.

39. In a pendulum experiment, L = 100.0 cm with an error of 0.1 cm and T = 2.00 s with an error of 0.02 s. The percentage error in g = 4π²L/T² is

    1. 3.1 %
    2. 1.1 %
    3. 2.1 %
    4. 1.0 %
    Answer

    C. 2.1 %

    (0.1/100.0) × 100 = 0.1 %; (0.02/2.00) × 100 = 1 %, doubled gives 2 %. Total 0.1 + 2 = 2.1 %.

40. The mass of a cube is 50.0 g with an error of 0.5 g. Its side is 2.00 cm with an error of 0.02 cm. The density with its error is

    1. 6.25 ± 0.50 g cm⁻³
    2. 6.25 ± 0.13 g cm⁻³
    3. 6.25 ± 0.08 g cm⁻³
    4. 6.25 ± 0.25 g cm⁻³
    Answer

    D. 6.25 ± 0.25 g cm⁻³

    ρ = 50.0/8.00 = 6.25. Percentage error = 1 + 3(1) = 4 %, so Δρ = 0.04 × 6.25 = 0.25.

41. The speed v of a transverse wave on a stretched string depends on the tension T and the mass per unit length μ. Using dimensions, v is proportional to

    1. √(Tμ)
    2. √(T/μ)
    3. T/μ
    4. T μ
    Answer

    B. √(T/μ)

    v = k Tᵃ μᵇ; [L T⁻¹] = [M L T⁻²]ᵃ [M L⁻¹]ᵇ. Mass: a + b = 0; time: −2a = −1, a = ½, b = −½. So v ∝ √(T/μ); check length: ½ + ½ = 1.

42. The potential difference across a resistor is V = 100 ± 5 V and the current is I = 10 ± 0.2 A. The resistance R = V/I is

    1. 10 ± 0.7 Ω
    2. 10 ± 0.3 Ω
    3. 10 ± 0.2 Ω
    4. 10 ± 0.5 Ω
    Answer

    A. 10 ± 0.7 Ω

    Percentage error = 5 % + 2 % = 7 %, so ΔR = 0.07 × 10 = 0.7 Ω.

43. A screw gauge has pitch 1 mm and 100 circular scale divisions. It shows a zero error of +0.05 mm. For a wire the main scale reads 3 mm and the circular scale reading is 47. The true diameter is

    1. 3.52 mm
    2. 3.42 mm
    3. 3.47 mm
    4. 3.37 mm
    Answer

    B. 3.42 mm

    Least count = 0.01 mm. Observed = 3 + 47 × 0.01 = 3.47 mm. A positive zero error is subtracted: 3.47 − 0.05 = 3.42 mm.