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APEX CLASS ESTD 2018

IITJEE: Analysis & Propagation Visualizer

By

Ashish Kumar Bahekar

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in

Apex class • IITJEE Units & Measurements Lab

Dimensional Analysis & Error Propagation Visualizer

Switch between the Percentage Error Lab and Dimensional Exponent Visualizer below!

🎯 Tap a JEE Formula to See How Individual Errors Combine:
Given: Mass Error (Δm/m) = 2% | Velocity Error (Δv/v) = 3% | Constant (½) Error = 0% Constant (½) 0% Error! + 1 × (Δm/m) 1 × 2% = 2% + 2 × (Δv/v) [Power = 2] 2 × 3% = 6% = MAX ΔK/K 8% Error Share Bar: Mass: 2% (25% share) Velocity²: 6% (75% of total error comes from v!) 🎯 JEE Rule: Pure numbers (½, 4π², 2) have ZERO error; exponents multiply the percentage error! Given: Length Error (ΔL/L) = 1% | Time Period Error (ΔT/T) = 2% | Find Max % Error in g Constant (4π²) 0% Error + 1 × (ΔL/L) 1 × 1% = 1% + + 2 × (ΔT/T) [Denominator!] 2 × 2% = 4% = MAX Δg/g 5% Error Share Bar: ΔL/L: 1% 2(ΔT/T): 4% (Added even though T² is in denominator!) ⚠️ #1 JEE Trap: Even though g = 4π²L × T⁻², we NEVER do 1% – 4%! Worst-case errors ALWAYS ADD (1% + 4% = 5%)! Given: Mass Error (ΔM/M) = 1.5% | Edge Length Error (ΔL/L) = 2.0% | Find Max % Error in Density (ρ) Mass Term: 1 × (ΔM/M) 1 × 1.5% = 1.5% + Volume Term (L³): 3 × (ΔL/L) 3 × 2.0% = 6.0% = MAX Δρ/ρ 7.5% Error Share Bar: Mass: 1.5% Volume (3 × ΔL/L): 6.0% (Cube triples the length error!) 💡 Takeaway: For a sphere or cube, radius/length error gets multiplied by 3 in Volume and by 2 in Surface Area! JEE PYQ: Z = A²B¹/² / (C¹/³D³) | % Errors in A, B, C, D are 1%, 2%, 3%, 4% respectively 2 × (ΔA/A) 2 × 1% = 2% + ½ × (ΔB/B) ½ × 2% = 1% + ⅓ × (ΔC/C) ⅓ × 3% = 1% + 3 × (ΔD/D) 3 × 4% = 12% = TOTAL ΔZ/Z 16% A: 2% B: 1% C: 1% D³ contributes 12% out of 16% (Measure D most carefully!) 🏆 Master Formula: ΔZ/Z = 2(ΔA/A) + ½(ΔB/B) + ⅓(ΔC/C) + 3(ΔD/D) = 2% + 1% + 1% + 12% = 16%!
Selected Physical Law
K = ½ m v² g = 4π² L / T² ρ = M / L³ Z = A²B¹/² / C¹/³D³
Product / Quotient Form
Error Propagation Rule
ΔK/K = Δm/m + 2(Δv/v) Δg/g = ΔL/L + 2(ΔT/T) Δρ/ρ = ΔM/M + 3(ΔL/L) Σ |power| × (% error)
All Powers Become Positive!
Dominant Error Source
Velocity v² (6%) Time Period T² (4%) Length L³ (6.0%) Variable D³ (12.0%)
Highest Exponent Term
Max Percentage Error
8.0 % 5.0 % 7.5 % 16.0 %
Worst-Case Total Uncertainty
🧬 Select a Physical Quantity to Inspect Its [Mᵃ Lᵇ Tᶜ] Exponents:
+3 +2 +1 0 (Base) -1 -2 FORCE (F = m × a) → [M¹ L¹ T⁻²] (SI Unit: kg·m·s⁻² = Newton) M⁺¹ (Mass) L⁺¹ (Length) T⁻² (Time⁻²) 🔑 Tip: Memorize Force [M¹L¹T⁻²] by heart—90% of Mechanics & Electrodynamics dimensions come directly from F! WORK / ENERGY / TORQUE (W = F × d) → [M¹ L² T⁻²] (SI Unit: Joule / N·m) M⁺¹ (Mass) L⁺² (Length²) T⁻² (Time⁻²) 🎯 JEE Favorite: Work, Kinetic Energy, Potential Energy, Heat, and Torque ALL share [M¹L²T⁻²]! COEFFICIENT OF VISCOSITY (η = F / 6πrv) → [M¹ L⁻¹ T⁻¹] (SI Unit: Pa·s or decapoise) M⁺¹ (Mass) L⁻¹ (Length⁻¹) T⁻¹ (Time⁻¹) ⚡ Fast Derivation: From Stokes’ Law F = 6πηνr → [η] = [MLT⁻²] / ([L] × [LT⁻¹]) = [M¹L⁻¹T⁻¹]! GRAVITATIONAL CONSTANT (G = F·r² / m₁m₂) → [M⁻¹ L³ T⁻²] (SI: N·m²·kg⁻²) M⁻¹ (Mass⁻¹) L⁺³ T⁻² (Time⁻²) 🌌 Notice Mass has a NEGATIVE power (-1) because m₁m₂ [M²] sits in the denominator!
Defining Formula
F = m × a W = F × d F = 6π η r v G = F r² / m²
Fastest Exam Recall
Dimensional Formula
[M¹ L¹ T⁻²] [M¹ L² T⁻²] [M¹ L⁻¹ T⁻¹] [M⁻¹ L³ T⁻²]
Exponents (a, b, c)
Identical Twins in JEE
Weight, Tension, Thrust Torque, Heat, Energy Stress × Time (P × t) g × L² / M
Same Dimensions
🔥 2-Second JEE Homogeneity Rule: Quantities can ONLY be added or subtracted if their dimensions are identical! Inside sin(θ), cos(ωt – kx), e⁻ᵃᵗ, or ln(bx), the argument inside the bracket is ALWAYS Dimensionless [M⁰L⁰T⁰] (so [ωt] = 1 and [kx] = 1).

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