Units and Measurements – Notes, MCQs, Quiz & Worksheet

Assuming v = k * g^a * R^b. Dimensions: [v] = L T^-1, [g] = L T^-2, [R] = L. Equating powers of L and T: L T^-1 = (L T^-2)^a * L^b => L^(a+b) T^(-2a). So, -2a = -1 => a = 1/2. And a+b = 1 => 1/2 + b = 1 => b = 1/2. Thus v = k * (gR)¹/2.

A: Angular momentum = M L² T^-1. Planck's constant = M L² T^-1. B: Stress = M L^-1 T^-2. Pressure = M L^-1 T^-2. C: Energy = M L² T^-2. Torque = M L² T^-2. So, All of the above is correct.

Energy [E] = M L² T^-2. Force [F] = M L T^-2. Acceleration [A] = L T^-2. From [F] = M [A], so M = [F][A]^-1. From [A] = L T^-2, so L = [A][T]². Substitute M and L into [E]: [E] = ([F][A]^-1) ([A][T]²)² [T]^-2 = [F][A]^-1 [A]² [T]⁴ [T]^-2 = [F A T²].

An Angstrom (1 Å = 10^-10 m) is a convenient unit because typical atomic radii are on the order of 0.5 Å to 2.5 Å. While nanometers (10^-9 m) and picometers (10^-12 m) are also SI-derived units, using Angstrom avoids small decimal numbers (like 0.05 nm) or large powers of 10 (like 50 pm) in everyday discussion of atomic sizes, making it more intuitive and practical in fields like crystallography and chemistry.

Assertion (A) is true (e.g., 2005 has 4 significant figures). Reason (R) is also true (e.g., 2500 has 2 significant figures). However, the reason for all zeros between non-zero digits being significant is not explained by the rule for trailing zeros. These are two separate rules for identifying significant figures. Hence, R is true but not the correct explanation for A.

Let f = k * l^a * T^b * m^c. Dimensions: f=[T^-1], l=[L], T=[M L T^-2], m=[M L^-1]. Substituting dimensions: [T^-1] = [L]^a * [M L T^-2]^b * [M L^-1]^c. Equating powers of M, L, T: M: 0 = b + c; L: 0 = a + b - c; T: -1 = -2b. From T: b = 1/2. From M: c = -b = -1/2. From L: a = c - b = -1/2 - 1/2 = -1. So, f = k * l^-1 * T^(1/2) * m^(-1/2) = (k/l) * sqrt(T/m).

The SI system is a coherent and rational system of units, meaning that derived units are obtained from fundamental units without involving numerical factors (like 1/2 or π) other than unity. It is also a decimal system, simplifying conversions, and covers all fields of science and engineering, promoting global consistency and ease of communication.

When resistances are connected in series, the equivalent resistance R = R1 + R2. So, R = 100 Ω + 150 Ω = 250 Ω. For addition (or subtraction), the absolute errors add up: ΔR = ΔR1 + ΔR2 = 3 Ω + 2 Ω = 5 Ω. So, the equivalent resistance is (250 ± 5) Ω.

Gram (mass), Second (time), and Centimeter (length) are fundamental units in the CGS system. Dyne is the CGS unit of force, which is a derived quantity (Force = Mass × Acceleration), hence it is a derived unit.

Least Count (LC) = Pitch / Number of circular scale divisions. Given LC = 0.01 mm and number of divisions = 100. So, Pitch = LC * Number of divisions = 0.01 mm * 100 = 1 mm.

Dimensional analysis cannot determine dimensionless constants (like 1/2, π, etc.) or constants that depend on the system of units used. It also cannot distinguish between physical quantities that have the same dimensions (e.g., work and torque, or pressure and energy density), and it's difficult to apply to equations involving trigonometric, exponential, or logarithmic functions directly.

Angular momentum = [M L² T^-1]. Planck's constant (h) also has dimensions [M L² T^-1] (from E=hf). Work and Energy are [M L² T^-2]. Torque is [M L² T^-2] and Power is [M L² T^-3]. Impulse is [M L T^-1].

Given 1 MSD = 1 mm. 20 VSD = 19 MSD, so 1 VSD = 19/20 MSD = 0.95 mm. Least Count (LC) = 1 MSD - 1 VSD = 1 mm - 0.95 mm = 0.05 mm.

The quantity I² R represents electric power. Dimensions of current I = [A]. Dimensions of resistance R = [M L² T^-3 A^-2]. So, Q = I² R = [A²] * [M L² T^-3 A^-2] = [M L² T^-3].

Numbers are 3.456 (3 decimal places), 2.3 (1 decimal place), 123.45 (2 decimal places). When adding or subtracting, the result must be rounded to the same number of decimal places as the number with the fewest decimal places. Sum = 3.456 + 2.3 + 123.45 = 129.206. The number with the fewest decimal places is 2.3 (one decimal place). So, the sum should be rounded to one decimal place: 129.2.

Distance = 4.25 light-years = 4.25 * (9.46 x 10¹5 m) = 40.155 x 10¹5 m = 4.0155 x 10¹6 m. To convert to km, divide by 1000: 4.0155 x 10¹6 m / 10³ = 4.0155 x 10¹3 km. Rounding to 3 significant figures, the answer is 4.02 x 10¹3 km.

Mole measures the amount of substance based on the number of elementary entities, while mass measures the inertia or quantity of matter. They represent fundamentally different concepts. One mole of a substance has a definite number of particles (Avogadro's number), irrespective of the mass of those particles, while mass depends on the actual quantity of matter. For example, 1 mole of hydrogen has a different mass than 1 mole of oxygen, but both represent the same count of particles.

The measurements are precise because they are very close to each other (they cluster around 12.31 cm, with a small range of 0.07 cm). The average of the readings (12.3125 cm) is very close to the actual length of 12.30 cm. Thus, the measurements are also accurate.

A six-month interval ensures that the Earth is at opposite ends of its orbit around the Sun. This provides the largest possible baseline (diameter of Earth's orbit, approximately 2 astronomical units) for observation. A larger baseline results in a larger parallax angle, which is easier to measure accurately, thereby reducing the percentage error in the calculated distance to the star.

The prefix 'fermi' is synonymous with 'femtometer'. 1 fermi (fm) is defined as 1 x 10^-15 meters.

Density ρ = m/V. The percentage error in ρ is (Δρ/ρ)% = (Δm/m)% + (ΔV/V)%. (Δm/m)% = (0.5/500.0) * 100% = 0.1%. (ΔV/V)% = (0.2/25.0) * 100% = 0.8%. Maximum percentage error = 0.1% + 0.8% = 0.9%.

Least Count (LC) = Pitch / Number of divisions on circular scale = 1 mm / 100 = 0.01 mm. Total Reading = Main Scale Reading (MSR) + Circular Scale Reading (CSR) * LC = 2 mm + 45 * 0.01 mm = 2 mm + 0.45 mm = 2.45 mm.

Reading a scale from an angle causes parallax error, which is a type of systematic error. Systematic errors are reproducible inaccuracies that are consistently in the same direction and arise from the experimental setup, calibration, or methodology.

Magnetic field strength B (Tesla) has dimensions [M A^-1 T^-2]. Magnetic intensity H (Ampere/meter) has dimensions [A L^-1]. From B = μH, we have μ = B/H. Dimensions of μ = [M A^-1 T^-2] / [A L^-1] = [M L T^-2 A^-2].

Let Q = L^(1/2). The fractional error is given by ΔQ/Q = (1/2) * (ΔL/L). Multiplying by 100% on both sides: (ΔQ/Q)% = (1/2) * (ΔL/L)%. Given (ΔL/L)% = X%, so the percentage error in √(L) is X/2 %.

The SI unit of electric current is indeed Ampere. Ampere is one of the seven fundamental (base) units in the SI system, meaning it is not derived from other units. The reason correctly explains why it's the SI unit—because it's chosen as a fundamental unit.

Random errors are unpredictable variations in measurements that lead to results differing in a random way from the true value, often due to fluctuations in experimental conditions or observer's judgment. Example: Fluctuations in air currents affecting a delicate balance. Systematic errors are consistent, reproducible errors that cause measurements to deviate consistently in one direction from the true value. Example: An incorrectly calibrated thermometer consistently reading 0.5°C higher than the actual temperature.

Least Count (LC) = 1 MSD - 1 VSD. Given 20 VSD = 19 MSD, so 1 VSD = 19/20 MSD. LC = (1/20) MSD. Given 1 MSD = 0.5 mm. So, LC = (1/20) * 0.5 mm = 0.025 mm. Reading = MSR + VSC * LC = 10.0 mm + 15 * 0.025 mm = 10.0 mm + 0.375 mm = 10.375 mm.

Perimeter P = 2(L + B). P = 2 * (15.3 + 12.8) = 2 * 28.1 = 56.2 cm. For sum, absolute errors add up: ΔP = 2 * (ΔL + ΔB) = 2 * (0.1 + 0.2) = 2 * 0.3 = 0.6 cm. So, the perimeter is (56.2 ± 0.6) cm.

Dimensions of R (Range) = [L]. Dimensions of u² = [L T^-1]² = [L² T^-2]. Dimensions of sin(2θ) are dimensionless. Dimensions of g = [L T^-2]. Dimensions of Right Hand Side = [L² T^-2] / [L T^-2] = [L]. Since the dimensions of LHS ([L]) match the dimensions of RHS ([L]), the equation is dimensionally correct.