TOPIC: Units, Dimensions, SI System, and Dimensional Analysis & Applications
IIT-JEE MAINS PYQ Arena — Set 4
Units, Dimensions, SI System & Dimensional Analysis
SI System & Electrical Conductance
JEE Main 2024 (Apr Shift 2)
⏱ JEE Standard Pace: 120s
+4 / -1 Scale
Q1. In the SI system of units (M, L, T, A), what is the correct dimensional formula for Electrical Conductivity (σ), which is the reciprocal of electrical resistivity?
✔ Spot On! +4 Marks
✖ Check L’s power: [M⁻¹ L⁻² T³ A²] is Conductance (G = 1/R), whereas σ = G / L!
Concept & Solution:From microscopic Ohm’s Law J = σ E:• Current density [J] = [I / Area] = [A L⁻²]
• Electric field [E] = [Force / Charge] = [M L T⁻²] / [A T] = [M L T⁻³ A⁻¹]
• Therefore, [σ] = [J / E] = [M⁻¹ L⁻³ T³ A²] (SI Unit: S·m⁻¹ or Ω⁻¹·m⁻¹).
Question 1 of 15
Trigonometric & Logarithmic Homogeneity
JEE Main 2023 (Jan Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q2. The vertical displacement y of an oscillating system is given by y = α² log_e(β t) + γ sin(ω t), where y is in metres and t is in seconds. What is the dimensional formula of the combination (α β / γ)?
✔ Sharp Eye on α²! +4 Marks
✖ Notice that α is squared in the equation: [α²] = [L], so [α] = [L¹/²]!
Concept & Solution:• Argument of log_e is dimensionless: [β t] = 1 ⇒ [β] = [T⁻¹].
• Since [α²] = [y] = [L] ⇒ [α] = [L¹/²].
• Since [γ] = [y] = [L¹].
• Therefore, [α β / γ] = ([L¹/²][T⁻¹]) / [L¹] = [M⁰ L⁻¹/² T⁻¹].
Quantum Hall Resistance & SI System
JEE Main 2023 (Apr Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q3. If h is Planck’s constant and e is the elementary electronic charge, then the physical quantity (h / e²) has the same dimensional formula as:
✔ Von Klitzing Constant (R_K = h/e²)! +4 Marks
✖ Check the dimensions: [h] = [Energy × Time] and [e²] = [Charge²].
Concept & Solution:• [h / e²] = [M L² T⁻¹] / [A T]² = [M L² T⁻³ A⁻²].
• Since Power P = I²R ⇒ [R] = [M L² T⁻³] / [A²] = [M L² T⁻³ A⁻²] (Electrical Resistance).
Changing Fundamental Quantities
JEE Main 2024 (Jan Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q4. If Pressure (P), Velocity (V) and Time (T) are chosen as the fundamental physical quantities, then the dimensional formula for Force (F) is:
✔ Instant 3-Second Derivation! +4 Marks
✖ Recall: Force = Pressure × Area, and Length = Velocity × Time!
Concept & Solution:• Force = Pressure × Area = P × L²
• Since Length [L] = [Velocity × Time] = [V T], Area [L²] = [V² T²].
• Therefore, [Force] = [P¹ V² T²].
Dimensional Formula Deduction (Acoustics)
JEE Main 2022 (July Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q5. The frequency (f) of a stretched string vibrating in its fundamental mode depends on its vibrating length (l), tension (T), and mass per unit length (μ) as f ∝ lᵃ Tᵇ μᶜ. What are the values of (a, b, c)?
✔ Sonometer Law Verified! +4 Marks
✖ Check the powers of M and Time: Tension T has [M L T⁻²] and linear density μ has [M L⁻¹].
Concept & Solution:• [M⁰ L⁰ T⁻¹] = [L]ᵃ [M L T⁻²]ᵇ [M L⁻¹]ᶜ = M^(b+c) L^(a+b−c) T^(−2b)
• Power of Time: −2b = −1 ⇒ b = 1/2
• Power of Mass: b + c = 0 ⇒ c = −1/2
• Power of Length: a + 1/2 − (−1/2) = 0 ⇒ a = −1. (Matches f = (1/2l)√(T/μ)).
Bohr Magneton & Magnetic Moment
JEE Main 2023 (Jan Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q6. In atomic physics, the Bohr Magneton is defined as μ_B = (e h) / (4π m_e), where e is electronic charge, h is Planck’s constant, and m_e is mass of an electron. What is the dimensional formula of μ_B?
✔ Magnetic Dipole Moment (I × Area)! +4 Marks
✖ Notice how T¹ in charge [A T] cancels T⁻¹ in Planck’s constant [M L² T⁻¹]!
Concept & Solution:• Bohr Magneton represents the fundamental quantum of Magnetic Dipole Moment (M = I × A):
• [μ_B] = ([A T] × [M L² T⁻¹]) / [M] = [M⁰ L² T⁰ A¹] (SI Unit: A·m² or J/T).
Poiseuille’s Law & Error Propagation
JEE Main 2024 (Jan Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q7. In an experiment to determine the coefficient of viscosity η = (π ΔP r⁴) / (8 Q l) of a liquid flowing through a capillary tube, the percentage errors in ΔP, r, Q, and l are 1%, 0.5%, 1%, and 1.5% respectively. What is the maximum percentage error in η?
✔ Precision Error Analysis! +4 Marks
✖ Watch out for r⁴: its percentage error gets multiplied by 4!
Concept & Solution:• (Δη / η) × 100 = (ΔP/P)% + 4(Δr/r)% + (ΔQ/Q)% + (Δl/l)%
• Maximum % Error = 1% + 4(0.5%) + 1% + 1.5% = 1% + 2% + 1% + 1.5% = 5.5%.
Planck Time & Cosmological Constants
JEE Main 2022 (June Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q8. Which of the following combinations of Universal Gravitational constant (G), Planck’s constant (h), and speed of light (c) has the dimension of Time [T]?
✔ Planck Time (t_P = l_P / c)! +4 Marks
✖ Remember: √(hG/c³) is Planck Length [L], so dividing by c inside the root gives c⁵!
Concept & Solution:• [h G] = [M¹ L² T⁻¹][M⁻¹ L³ T⁻²] = [L⁵ T⁻³].
• Dividing by [c⁵] = [L⁵ T⁻⁵] gives [h G / c⁵] = [T²].
• Taking the square root gives √(h G / c⁵) = [T¹] (Planck Time ≈ 5.39 × 10⁻⁴⁴ s).
SI System & Damped Oscillations
JEE Main 2023 (Apr Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q9. The differential equation of a damped harmonic oscillator is given by m (d²x / dt²) + b (dx / dt) + k x = 0. What is the dimensional formula of the ratio (b² / m k)?
✔ Dimensionless Damping Ratio Squared! +4 Marks
✖ Derive [b] and [k] from Force: [b] = [M T⁻¹] and [k] = [M T⁻²].
Concept & Solution:• Damping coefficient: [b] = [Force / Velocity] = [M L T⁻²] / [L T⁻¹] = [M¹ T⁻¹].
• Spring constant: [k] = [Force / Length] = [M¹ T⁻²].
• Therefore, [b² / (m k)] = [M² T⁻²] / ([M¹][M¹ T⁻²]) = [M⁰ L⁰ T⁰].
SI System & Elasticity
JEE Main 2021 (Mar Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q10. Compressibility (β) of a fluid is defined as the fractional change in volume per unit increase in pressure, i.e., β = −(1 / V)(dV / dP). What is the dimensional formula of compressibility (β)?
✔ Reciprocal of Bulk Modulus! +4 Marks
✖ Remember: Pressure has [M¹ L⁻¹ T⁻²], so its reciprocal flips all three signs!
Concept & Solution:• Volumetric strain (dV / V) is dimensionless.
• Therefore, [β] = 1 / [dP] = 1 / [M¹ L⁻¹ T⁻²] = [M⁻¹ L¹ T²] (SI Unit: Pa⁻¹ or m²/N).
Conversion of System of Units (G)
JEE Main 2024 (Apr Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q11. The value of the Universal Gravitational Constant in CGS units is 6.67 × 10⁻⁸ dyne·cm²·g⁻². If a new system of units uses 100 g as the unit of mass, 10 m as the unit of length, and 100 s as the unit of time, what will be the numerical value of G in this new system?
✔ Flawless Exponent Tracking! +4 Marks
✖ Convert 10 m to 10³ cm so both systems are compared in CGS base!
Concept & Solution: Since [G] = [M⁻¹ L³ T⁻²]:• n₂ = 6.67×10⁻⁸ × (1 g / 10² g)⁻¹ × (1 cm / 10³ cm)³ × (1 s / 10² s)⁻²
• n₂ = 6.67×10⁻⁸ × (10²) × (10⁻⁹) × (10⁴) = 6.67×10⁻⁸ × 10⁻⁵ = 6.67 × 10⁻¹³.
SI System & Rotational Dynamics
JEE Main 2022 (July Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q12. In rotational mechanics, the restoring torque (τ) per unit twist angle (θ) of a suspension wire is called the Torsional Constant (C = τ / θ). What is the ratio of the dimensional formula of Torsional Constant (C) to Moment of Inertia (I)?
✔ Torsional Pendulum ω² = C / I! +4 Marks
✖ Recall the time period of a torsional pendulum: T = 2π √(I / C)!
Concept & Solution:• Since θ is dimensionless, Torsional Constant [C] = [τ] = [M¹ L² T⁻²].
• Moment of Inertia [I] = [M¹ L² T⁰].
• Ratio [C / I] = [M¹ L² T⁻²] / [M¹ L² T⁰] = [M⁰ L⁰ T⁻²] (Angular frequency squared, ω²).
Thermodynamics & SI Units
JEE Main 2023 (Jan Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q13. Which of the following pairs of thermal quantities has the exact same dimensional formula [M¹ L² T⁻² K⁻¹]?
✔ Both are Energy / Temperature! +4 Marks
✖ Note: Universal Gas Constant R has [mol⁻¹] in the denominator, and Specific Heat has [M⁰]!
Concept & Solution:• Entropy: dS = dQ / T ⇒ [S] = [Energy / Temperature] = [M¹ L² T⁻² K⁻¹].
• Boltzmann Constant: E = (3/2) k_B T ⇒ [k_B] = [Energy / Temperature] = [M¹ L² T⁻² K⁻¹] (J/K).
SI System & Maxwell’s Displacement Current
JEE Main 2024 (Jan Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q14. In the Ampere-Maxwell law ∮ B · dl = μ₀ (I_c + ε₀ dΦ_E / dt), where Φ_E is electric flux and ε₀ is permittivity of free space, what is the dimensional formula of the term ε₀ (dΦ_E / dt)?
✔ Principle of Homogeneity makes it instant! +4 Marks
✖ Since ε₀(dΦ_E/dt) is added directly to conduction current I_c, it must have the dimension of Current [A]!
Concept & Solution:• The term I_d = ε₀ (dΦ_E / dt) is Maxwell’s Displacement Current.
• By the Principle of Homogeneity, since it is added directly to conduction current I_c, its dimensional formula is simply [M⁰ L⁰ T⁰ A¹].
Blast Wave Dimensional Analysis (Taylor-Sedov)
JEE Main 2023 (Apr Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q15. The radius (R) of a spherical shockwave expanding from a point explosion depends on the energy (E) released, the ambient air density (ρ), and the elapsed time (t) as R ∝ Eᵃ ρᵇ tᶜ. Find the values of (a, b, c):
✔ Legendary G.I. Taylor Blast Wave Formula! +4 Marks
✖ Match powers of M first (a + b = 0), then L (2a − 3b = 1) to get 5a = 1!
Concept & Solution:• [M⁰ L¹ T⁰] = [M¹ L² T⁻²]ᵃ [M¹ L⁻³]ᵇ [T¹]ᶜ = M^(a+b) L^(2a−3b) T^(−2a+c)
• Power of M: a + b = 0 ⇒ b = −a
• Power of L: 2a − 3(−a) = 1 ⇒ 5a = 1 ⇒ a = 1/5 and b = −1/5
• Power of T: −2(1/5) + c = 0 ⇒ c = 2/5. Thus R ∝ (E t² / ρ)^(1/5).
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