Units & measurements practice · Mixed difficulty

Option A is correct because Statement I is true, Statement II is true, and Statement II is the correct explanation for Statement I. Dimensional analysis cannot determine the dependence on dimensionless quantities like angles.

Option A ([M L² T⁻²]) is correct because torque is force times distance, and the dimensions of force are [M L T⁻²] and distance is [L], so torque is [M L² T⁻²].

Option A (Radius of gyration and light year) is correct because both radius of gyration and light year have the dimension of length.

The correct answer is C ($M{L^{ - 1}}{T^{ - 1}}$) because the coefficient of viscosity is defined by the formula F = -ηA(dv/dx), and solving for η gives the stated dimensions.

The correct answer is Magnetic flux because its dimensions are equivalent to inductance multiplied by current, which matches the given dimensions.

Option A (${10^3}$) is correct because the ratio of the SI unit to the CGS unit of the universal gravitational constant is calculated using dimensional analysis.

Option B ($q{v^2}$) is correct because Farad is the unit of capacitance, which is equivalent to $\frac{q}{V}$, $\frac{{{q^2}}}{J}$, and $\frac{J}{{{V^2}}}$, but not $q{v^2}$.

The correct answer is $\frac{\alpha \gamma}{\beta}$ because the dimensions of each term in the potential energy equation must be the same, allowing us to find a dimensionless combination.

Option C is correct because Statement-I is false and Statement-II is true. The error in kinetic energy calculation is incorrect, but the formula for error propagation is correct.

Option B is correct because plane angle is defined as the ratio of arc length to radius (both lengths), making it unitless and dimensionless, and all unitless quantities are dimensionless.

Option A ($M{L^2}{T^{ - 2}}$) is correct because heat energy, like all forms of energy, has the dimensions of mass times length squared divided by time squared.

Option D (Is reduced by a factor of 1/4) is correct because the error is inversely proportional to the square root of the number of readings. Since the number of readings is multiplied by 4, the error is divided by the square root of 4, which is 2. The variance is reduced by a factor of 4.

Option A is correct because it accurately counts the significant figures in each given measurement based on the rules of significant figures.

Option B ([${M^0}L{T^0}K$]) is correct because Wien's constant has dimensions of length times temperature.

Option C (0.1) is correct because the conversion factor from CGS to MKS for Young's modulus (dyne/cm²) to N/m² is 0.1.

Option A is correct because force has the dimensional formula [MLT⁻²], while pressure, stress, and modulus of elasticity have [ML⁻¹T⁻²].

Option C (Newton/metre${ }^2$) is correct because Young's modulus is defined as stress/strain, where stress has units of force/area (N/m²) and strain is dimensionless.

Option A (Pressure) is correct because Young's modulus is defined as stress/strain, and stress has the same units as pressure.

Option D ($n \propto \frac{1}{u}$) is correct because the physical quantity is constant, implying an inverse relationship between numerical value and unit.

Option C (Weber) is correct because the Weber is the SI unit of magnetic flux.

Option D (4%) is correct because the maximum percentage error in density is the sum of the percentage errors in mass, length, and twice the percentage error in radius.

Option D ([${L}^{-1} {~A}$]) is correct because the intensity of magnetization is defined as the magnetic moment per unit volume, which has the dimensions of current per unit length.

Option D (129600) is correct because the acceleration is transformed using the conversion factors for length and time to the new system of units.

Option A is correct because by equating dimensions of each term in the given equation, we can find the dimensions of a, b, c, and d, and then calculate the dimensions of ab/cd.

Option C ($Joule \\;{K^{ - 1}}mo{l^{ - 1}}$) is correct because the universal gas constant (R) is expressed in terms of energy per mole per Kelvin.

Option C is correct because both the assertion and reason are true, and the reason correctly explains how the least count of a screw gauge is calculated.

Option B (${c^0}{g^2}{p^{ - 1}}$) is correct because it is derived by equating the dimensions of the gravitational constant with the given fundamental quantities.

Option B (${M^{ - 1}}{L^2}T$) is correct because by equating dimensions, we find the dimension of 'a' and 'b' and then multiply them to get the dimension of a x b.

Option A (1 mm) is correct because the pitch of a screw gauge is the distance it moves forward in one complete revolution, which is given as 1 mm.

Option D (Newton/ coulomb) is correct because electric intensity (electric field) is defined as force per unit charge.