Physics and Measurement

Dimensions are the powers of base quantities (M, L, T, A, K, mol, cd) in a derived quantity.

What dimensional analysis CAN do:

1. Check correctness of an equation — both sides must have the same dimensions.
2. Convert units between systems (e.g. SI to CGS).
3. Derive relationships between physical quantities up to a dimensionless constant.

What it CANNOT do:

1. Find the dimensionless constants (like 2π in T = 2π√(L/g)).
2. Distinguish between quantities that share dimensions (work and torque are both [ML²T⁻²]).
3. Handle equations involving trigonometric, exponential or log functions of dimensional quantities (their arguments must be dimensionless).

Worked example. Estimate the time period T of a simple pendulum, given it depends on length L, mass m, and g. Assume T = k · L^a · m^b · g^c. Equating dimensions: [T] = [L]^a [M]^b [LT⁻²]^c. So 1 = -2c → c = -1/2. 0 = a + c → a = 1/2. 0 = b → b = 0. Result: T ∝ √(L/g). The constant 2π comes from full Newtonian mechanics, not from dimensions.

Significant figures (sig figs) express measurement precision. Rules: (1) All non-zero digits are significant. (2) Zeros between non-zeros are significant (e.g. 1002 has 4). (3) Leading zeros are NOT significant (0.0025 has 2). (4) Trailing zeros after a decimal point ARE significant (2.300 has 4). (5) Trailing zeros in a whole number without a decimal are ambiguous (1500 is treated as 2 sig figs unless written 1.500x10^3). Memory aid: 'Sandwiched zeros count, leading zeros don't, trailing-after-decimal do.' Scientific notation removes ambiguity: only the mantissa digits count. In JEE, exact constants (like the 2 in 2pi) have infinite sig figs and never limit precision. Always identify sig figs before applying rounding rules in numerical problems.

For addition/subtraction (Z = A +/- B), absolute errors ADD: deltaZ = deltaA + deltaB. For multiplication/division (Z = AB or A/B), relative (fractional) errors ADD: deltaZ/Z = deltaA/A + deltaB/B. For powers Z = A^p B^q / C^r: deltaZ/Z = |p|(deltaA/A) + |q|(deltaB/B) + |r|(deltaC/C). Key shortcut: powers multiply the fractional error by the exponent, so a quantity raised to a high power dominates total error. Maximum permissible error always uses the SUM of magnitudes (worst case), never subtraction. Percentage error = (deltaZ/Z)x100. Result of add/subtract is rounded to least decimal places; result of multiply/divide is rounded to least significant figures. These formulas are among the most frequently tested in the Measurement chapter.

A cube has mass m = 10.0 g (deltam = 0.1 g) and side L = 2.00 cm (deltaL = 0.01 cm). Find density rho = m/V = m/L^3 and its percentage error. Compute rho = 10.0/(2.00)^3 = 10.0/8.00 = 1.25 g/cm^3. Error: since rho = m L^-3, deltarho/rho = deltam/m + 3(deltaL/L) = (0.1/10.0) + 3(0.01/2.00) = 0.01 + 3(0.005) = 0.01 + 0.015 = 0.025. So percentage error = 2.5%. Note the side's error is multiplied by 3 (the power), making it the larger contributor even though deltaL/L is small. Final: rho = 1.25 g/cm^3 with 2.5% error, i.e. deltarho = 0.03 g/cm^3. This power-multiplication step is the classic JEE trap.