Dimensional Analysis — Finding dimensions of quantities
Units & Measurements
10
JEE Qs
8%
Hard
60
min
Master the derivation of dimensions for all common physical quantities and constants by applying the principle of homogeneity systematically to their defining formulas.
🧮 Key Formulas
✅ Key Points for JEE
- 1Dimensions of a physical quantity are expressed as a product of powers of the seven fundamental dimensions: Mass (M), Length (L), Time (T), Electric Current (I), Thermodynamic Temperature (θ), Amount of Substance (N), and Luminous Intensity (J).
- 2According to the principle of homogeneity, only physical quantities having the same dimensions can be added, subtracted, or equated; multiplication and division are not restricted by dimensions.
- 3Dimensionless quantities (e.g., pure numbers, trigonometric ratios, logarithmic functions, exponential terms, strain, refractive index) always have dimensions [M^0 L^0 T^0].
- 4To find the dimensions of a derived quantity, substitute the dimensions of the constituent fundamental quantities into its defining formula and simplify the powers of M, L, T, etc.
- 5All physical constants (e.g., gravitational constant G, Planck's constant h, permittivity ε₀, permeability μ₀) have specific dimensions, which are derived from their respective defining equations.
⚠️ Common Mistakes
- ✕Confusing units with dimensions; for example, using 'kg' instead of 'M' or 'm' instead of 'L' in dimensional formulas.
- ✕Incorrectly identifying the fundamental formula for a given quantity, leading to an incorrect dimensional derivation.
- ✕Failing to apply the principle of homogeneity correctly, especially in arguments of trigonometric, exponential, or logarithmic functions, which must always be dimensionless.
- ✕Algebraic errors while combining exponents of M, L, T, etc., during the simplification process.
📝 Practice Questions
See allQ43.The maximum percentage error in the measurment of density of a wire is [Given, mass of wire = (0.60 ± 0.003)g radius of wire = (0.50 ± 0.01)cm length of wire = (10.00 ± 0.05)cm] (1) 8 (2) 5 (3) 4 (4) 7 for another diatomic molecules, but for rigid molecules and γ2 =
Q43.The energy of a system is given as E(t) = α3e−βt , where t is the time and β = 0.3 s−1 . The errors in the measurement of α and t are 1.2% and 1.6% , respectively. At t = 5 s, maximum percentage error in the energy is : (1) 6% (2) 8.4% (3) 11.6% (4) 4%
Q39.Given below are two statements : Statement I : In a vernier callipers, one vernier scale division is always smaller than one main scale division. Statement II : The vernier constant is given by one main scale division multiplied by the number of vernier scale divisions. In the light of the above statements, choose the correct answer from the options given below. (1) Statement I is true but Statement II is false (2) Statement I is false but Statement II is true (3) Both Statement I and Statement II are false (4) Both Statement I and Statement II are true 2025 (22 Jan Shift 1) JEE Main Previous Year Paper
Q40.Which one of the following is the correct dimensional formula for the capacitance in F ? M, L, T and C stand for unit of mass, length, time and charge, (1) [F] = [C2M−1 L−2 T2] (2) [F] = [C2M−2 L2 T2] (3) [F] = [CM−2 L−2 T−2] (4) [F] = [CM−1 L−2 T2]
Q28.The position of a particle moving on x-axis is given by x(t) = A sin t + B cos2 t + Ct2 + D , where t is time. The dimension of ABC is D (1) L2T −2 (2) L2 (3) L (4) L3 T−2
Q45.For an experimental expression y = 32.3×112527.4 , where all the digits are significant. Then to report the value of y we should write (1) y = 1326.19 (2) y = 1330 (3) y = 1326.186 (4) y = 1326.2
NCERT Chapters
- Class 11 Physics Ch 2: Units and Measurements