MechanicsHigh Yield (1-2 Qs in JEE)

Units, Dimensions & Errors Formulas

Dimensional analysis, dimensional formulas of physical quantities, propagation of errors, vernier callipers and screw gauge.

8 Formulas•462+ Questions in Bank

Core Formulas & Mathematical Identities

Z = \frac{A^p B^q}{C^r} \implies \frac{\Delta Z}{Z} = p \frac{\Delta A}{A} + q \frac{\Delta B}{B} + r \frac{\Delta C}{C}

Fractional and percentage error estimation in derived quantities.

Examiner Tip: Errors always add up; never subtract error terms even if quantity is in the denominator.

\text{LC}_{\text{vernier}} = 1 \text{ MSD} - 1 \text{ VSD} = \frac{1 \text{ MSD}}{N}, \quad \text{LC}_{\text{screw gauge}} = \frac{\text{Pitch}}{\text{Total Circular Scale Divisions}}

Instrument precision formulas for laboratory physics where N divisions of vernier scale coincide with (N - 1) main scale divisions.

Examiner Tip: Pitch is distance advanced in one complete rotation of the thimble (usually 0.5 mm or 1.0 mm).

\text{True Reading} = \text{Observed Reading} - (\text{Zero Error}), \quad \text{Observed Reading} = \text{MSR} + (\text{VSR} \times \text{LC})

Correction for instrumental zero offset. If zero is to the right/above, error is positive; if to the left/below, error is negative.

Examiner Tip: Positive zero error is SUBTRACTED; negative zero error is ADDED (since -(-e) = +e).

[G] = [M^{-1}L^3T^{-2}], \quad [h] = [ML^2T^{-1}], \quad [\varepsilon_0] = [M^{-1}L^{-3}T^4A^2], \quad [\mu_0] = [MLT^{-2}A^{-2}]

Fundamental universal constants tested in JEE Main dimensional matching.

Examiner Tip: Speed of light relation: c = 1 / √(μ₀ ε₀) has dimension [LT⁻¹].

\text{For } \theta \ll 1 \text{ rad}: \sin\theta \approx \theta, \quad \tan\theta \approx \theta, \quad \cos\theta \approx 1 - \frac{\theta^2}{2}; \quad (1 + x)^n \approx 1 + nx \quad (|x| \ll 1)

Essential mathematical approximations for small oscillations, optics angles, thermal expansions, and gravitation height variations.

Examiner Tip: Angle θ must be converted to radians before applying small angle trigonometric approximations.

\text{Add/Sub: Least decimal places}; \quad \text{Mult/Div: Least significant figures}; \quad \sigma_{\text{mean}} = \frac{\sigma}{\sqrt{N}}

Precision rules for experimental physics observations and standard deviation of sample mean.

Examiner Tip: All non-zero digits are significant; zeroes between non-zero digits are significant; trailing zeroes after decimal are significant.

Z = \frac{A^p B^q}{C^r} \implies \frac{\Delta Z}{Z} = p\left(\frac{\Delta A}{A}\right) + q\left(\frac{\Delta B}{B}\right) + r\left(\frac{\Delta C}{C}\right)

Logarithmic differentiation error combination. Fractional and percentage errors add up in quadrature or worst-case linear bound.

Examiner Tip: In addition or subtraction (Z = A ± B), absolute errors always add: ΔZ = ΔA + ΔB (never subtract errors).

R = \frac{l^2}{6h} + \frac{h}{2}, \quad \text{Pitch} = \frac{\text{Distance moved}}{\text{Rotations}}, \quad \text{LC} = \frac{\text{Pitch}}{N}

Laboratory measurement of radius of curvature of convex/concave optical lenses where l is mean leg separation and h is central sagitta elevation.

Examiner Tip: For very small sagitta (h << l), R ≈ l² / (6h).

Frequently tested misconceptions in Units, Dimensions & Errors

•Pure numbers and trigonometric arguments are dimensionless [M⁰L⁰T⁰].
•Percentage error in quantity Z = A^p B^q / C^r is (ΔZ/Z)*100 = p(ΔA/A)*100 + q(ΔB/B)*100 + r(ΔC/C)*100 (errors ALWAYS ADD).