Questions & Answers: "units and measurement"

Using the rule for propagation of errors in products and powers: (ΔP/P) = 3(Δa/a) + 2(Δb/b) + (1/2)(Δc/c) + (Δd/d). Substituting the given percentage errors: % error in P = 3(1%) + 2(3%) + 0.5(4%) + 2% = 3% + 6% + 2% + 2% = 13%. Options 7%, 10%, and 14% are incorrect because they result from miscalculating the power coefficients or incorrectly applying the square root factor.

From Newton's law of gravitation, F = G m1 m2 / r². Thus, G = F r² / (m1 m2). Substituting dimensions: [G] = [MLT⁻²] [L²] / [M²] = [M⁻¹ L³ T⁻²]. The other options are incorrect as they do not balance the mass or length dimensions correctly according to the force equation.

In 0.007, trailing zeros after a decimal point but before a non-zero digit are not significant, so only '7' is significant (1 figure). In scientific notation (2.64 × 10²⁴), only the coefficients are counted, so '2', '6', and '4' are significant (3 figures). 4 and 3 is wrong because leading zeros are not significant. 3 and 3 is wrong because the zeros in 0.007 are not significant. 1 and 24 is wrong because the power of 10 does not affect significant figures.

Magnification = Observed size / Real size. Therefore, Real size = Observed size / Magnification = 3.5 mm / 100 = 0.035 mm. 3.5 mm is the observed size, not the actual size. 0.35 mm and 0.0035 mm result from incorrect division by 10 or 1000 respectively.

By the principle of homogeneity, dimensions of each term on the RHS must equal dimensions of the LHS. So, [bt²] = [x]. [b] [T²] = [L]. Thus, [b] = [L T⁻²]. The unit is meters per second squared (m/s²). m/s is the unit for 'a', m s and m are dimensionally inconsistent with the term bt² equaling a length.

Impulse = Force × Time = [MLT⁻²][T] = [MLT⁻¹]. Momentum = Mass × Velocity = [M][LT⁻¹] = [MLT⁻¹]. Work [ML²T⁻²] and Power [ML²T⁻³] are different. Stress [ML⁻¹T⁻²] and Strain [dimensionless] are different. Force [MLT⁻²] and Pressure [ML⁻¹T⁻²] are different.

Volume of a sphere V = (4/3)πr³. The relative error is ΔV/V = 3(Δr/r). Given Δr/r = 2%, the percentage error in volume is 3 × 2% = 6%. 2% is the error in radius, 4% would be the error in surface area (r²), and 8% is a calculation error.

Least Count (LC) = Pitch / No. of divisions = 1 mm / 100 = 0.01 mm. Diameter = Main Scale Reading + (Circular Scale Reading × LC) = 1 mm + (47 × 0.01 mm) = 1 + 0.47 = 1.47 mm. 1.047 mm and 1.0047 mm are errors in LC calculation. 1.47 cm uses incorrect units.

In multiplication and division, the result should have the same number of significant figures as the term with the least significant figures. 25.2 has 3, 1374 has 4, and 33.3 has 3. The result must have 3 significant figures. Calculation: 1039.78... Rounding to 3 sig figs gives 1040 (The zero is not significant unless specified). 1039.78 and 1039.8 have too many sig figs. 1000 has only 1 sig fig.

1 parsec is defined as the distance at which an arc of 1 AU length subtends an angle of 1 arc second. θ = l / r => 1" = 1 AU / 1 parsec. Since 1" = (1/3600) * (π/180) radians, 1 parsec = 1 AU / radians value ≈ 206265 AU. 3.08 × 10¹⁶ is the value in meters, not AU. 1.496 × 10¹¹ is the value of 1 AU in meters.

Solar constant = Energy / (Area × Time) = [ML²T⁻²] / ([L²][T]) = [ML⁰T⁻³]. [ML²T⁻²] is energy, [ML¹T⁻²] is force, and [ML⁰T⁻²] is surface tension/spring constant. These do not represent energy per area per time.

Let F = [P^a V^b T^c]. [MLT⁻²] = [ML⁻¹T⁻²]^a [LT⁻¹]^b [T]^c. Equating powers of M: a = 1. Equating powers of L: -a + b = 1 => -1 + b = 1 => b = 2. Equating powers of T: -2a - b + c = -2 => -2(1) - 2 + c = -2 => c = 2. So F = [P V² T²]. Other options fail the dimensional consistency check for mass, length, or time.

A light year is the distance travelled by light in vacuum in one year; it is a unit of distance ([L]), not time. Leap year, Lunar month, and Solar day are all measures of time intervals. This is a common trap in NCERT physics.

Least Count = 1 MSD - 1 VSD. Given 10 VSD = 9 MSD, so 1 VSD = 0.9 MSD. LC = 1 MSD - 0.9 MSD = 0.1 MSD. Since 1 MSD = 1 mm, LC = 0.1 mm. 0.01 mm is typical for screw gauges. 1 mm is the MSD value. 0.9 mm is the value of one VSD, not the least count.

Volume = length × breadth × thickness = 4.234 m × 1.005 m × 0.0201 m = 0.0855289... m³. Since the thickness (2.01 cm = 0.0201 m) has 3 significant figures, the result must be rounded to 3 significant figures, giving 0.0855 m³. 0.08553 and 0.085529 have too many significant figures. 0.086 has too few.