TOPIC: Units, Dimensions, SI System, and Dimensional Analysis & Applications
IIT-JEE MAINS PYQ Arena — Set 6
Units, Dimensions, SI System & Dimensional Analysis
SI System & Angular Frequency
JEE Main 2024 (Jan Shift 2)
⏱ JEE Standard Pace: 120s
+4 / -1 Scale
Q1. The angular frequency (ω) of a mass m suspended from a spring of spring constant k is given by ω = √(k / m). What is the dimensional formula of angular frequency ω?
✔ Basic Frequency Dimension! +4 Marks
✖ Reminder: Angular frequency has the exact same dimensions as regular frequency [T⁻¹].
Concept & Solution:• Spring constant [k] = [Force / Length] = [M¹ T⁻²]. Mass [m] = [M¹].
• [ω] = √([M¹ T⁻²] / [M¹]) = √[T⁻²] = [M⁰ L⁰ T⁻¹].
Question 1 of 15
Principle of Homogeneity
JEE Main 2024 (Apr Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q2. The acceleration a of a particle is given in terms of time t by a = p t + q / (t + r). What are the dimensional formulas of constants p and q respectively?
✔ Homogeneity Mastered! +4 Marks
✖ Check [pt] = [a] ⇒ [p] = [L T⁻²] / [T] = [L T⁻³].
Concept & Solution:• [p t] = [a] ⇒ [p] = [L T⁻²] / [T] = [L T⁻³].
• Since [r] = [t] = [T], denominator is [T].
• [q / T] = [a] ⇒ [q] = [L T⁻²] × [T] = [L T⁻¹].
SI System & Magnetic Dipole Moment
JEE Main 2023 (Jan Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q3. In the SI system of fundamental units, what is the correct dimensional formula for Magnetic Dipole Moment (M_m)?
✔ Current × Area! +4 Marks
✖ Recall definition: Magnetic Moment M = i × A (Current × Area).
Concept & Solution:• [M_m] = [Current] × [Area] = [A¹] × [L²] = [M⁰ L² T⁰ A¹] (SI Unit: A·m²).
Changing Fundamental Quantities
JEE Main 2023 (Apr Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q4. If Force (F), Acceleration (A), and Time (T) are chosen as the fundamental physical quantities, what will be the dimensional formula for Energy (E)?
✔ Fast Energy Derivation! +4 Marks
✖ Hint: Energy = Force × Displacement. Distance [L] = [Acceleration × Time²]!
Concept & Solution:• Energy = Force × Displacement = F × L
• Since Acceleration [A] = [L / T²] ⇒ Displacement [L] = [A T²].
• Therefore, [Energy] = [F¹ A¹ T²].
Dimensional Analysis in Hydrodynamics
JEE Main 2022 (July Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q5. The critical velocity (v_c) of liquid flow through a tube depends on fluid density (ρ), tube radius (r), and coefficient of viscosity (η) as v_c ∝ ρᵃ rᵇ ηᶜ. What are the values of (a, b, c)?
✔ Reynolds Number Basis! +4 Marks
✖ Use Reynolds number Re = ρ v r / η = constant to instantly find a = −1, b = −1, c = 1.
Concept & Solution:• Since Re is dimensionless: [ρ][v][r] / [η] = [M⁰ L⁰ T⁰] ⇒ v ∝ η / (ρ r).
• Therefore, a = −1, b = −1, c = 1.
Error Propagation in Kinetic Energy
JEE Main 2024 (Jan Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q6. The kinetic energy of a body is calculated using K = (1/2) m v². If the percentage errors in measuring mass m and velocity v are 1.5% and 2% respectively, what is the maximum percentage error in kinetic energy K?
✔ Velocity Error Doubled! +4 Marks
✖ Remember: Velocity is squared, so its percentage error (2%) is multiplied by 2!
Concept & Solution:• (ΔK / K) × 100 = (Δm / m)% + 2(Δv / v)%
• Maximum % Error = 1.5% + 2(2%) = 1.5% + 4% = 5.5%.
SI System & Radioactivity
JEE Main 2023 (Jan Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q7. According to radioactive decay law (dN/dt = −λ N), what is the dimensional formula of the decay constant (λ)?
✔ Inverse Time Constant! +4 Marks
✖ Notice that N is a pure dimensionless count, so λ has the dimensions of 1/Time.
Concept & Solution:• Since λ = (1 / N) (dN / dt) and N is pure number count:
• [λ] = 1 / [Time] = [M⁰ L⁰ T⁻¹] (SI Unit: s⁻¹).
Custom Unit Systems & Velocity
JEE Main 2022 (June Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q8. In a system of units where the unit of acceleration is 2 m/s² and the unit of force is 10 N, what is the new unit of Mass?
vn Newton’s Second Law M = F / a! +4 Marks
✖ Apply F = m a directly: m = F / a = 10 N / 2 m/s² = 5 kg.
Concept & Solution:• From F = m a ⇒ Mass = Force / Acceleration = 10 N / 2 m/s² = 5 kg.
SI System & Magnetic Energy
JEE Main 2023 (Apr Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q9. If L is inductance and I is current, the dimensional formula of stored magnetic energy (1/2 L I²) is:
✔ Standard Energy Dimension! +4 Marks
✖ Remember: Any form of stored energy (mechanical, electrostatic, magnetic) has dimensions [M L² T⁻²].
Concept & Solution:• Magnetic energy in an inductor is strictly Energy: [M¹ L² T⁻²] (Joules).
SI System & Thermodynamics
JEE Main 2024 (Apr Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q10. In van der Waals equation (P + a / V²)(V − b) = n R T, what is the dimensional formula of the product (P b)?
✔ Pressure × Volume = Energy! +4 Marks
✖ Notice that [b] = [Volume] = [L³], and [P] = [M L⁻¹ T⁻²]. Product [P b] = [M L² T⁻²].
Concept & Solution:• [P] = [M L⁻¹ T⁻²], [b] = [Volume] = [L³].
• Product [P b] = [M L⁻¹ T⁻²] × [L³] = [M¹ L² T⁻²] (matches Energy/Work).
Transcendental Homogeneity
JEE Main 2021 (Aug Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q11. The electric field in an electromagnetic wave is given by E = E_0 sin(kx − ωt). What is the dimensional formula of the product (E_0 k)?
vn Gradient of Electric Field! +4 Marks
✖ Check dimensions: Electric field [E_0] = [M L T⁻³ A⁻¹] and wave number [k] = [L⁻¹].
Concept & Solution:• Electric Field [E_0] = [M¹ L¹ T⁻³ A⁻¹].
• Wave number [k] = [L⁻¹].
• Product [E_0 k] = [M¹ L¹ T⁻³ A⁻¹] × [L⁻¹] = [M¹ L⁰ T⁻² A⁻¹] (Force per unit charge per unit length).
Conversion of Units (Density)
JEE Main 2024 (Jan Shift 2)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q12. The density of a material is 13.6 g/cm³. When expressed in SI units of kg/m³, its numerical value will be:
✔ Standard CGS-to-SI Conversion! +4 Marks
✖ Remember: 1 g/cm³ = 10³ kg / 10⁻⁶ m³ = 10³ kg/m³.
Concept & Solution:• 1 g/cm³ = 10³ kg/m³. Therefore, 13.6 g/cm³ = 13.6 × 10³ = 1.36 × 10⁴ kg/m³.
Dimensional Analysis in Gravitation
JEE Main 2023 (Jan Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q13. The escape velocity (v_e) from a planet of mass M and radius R depends on gravitational constant G as v_e ∝ Mᵃ Rᵇ Gᶜ. What are the values of (a, b, c)?
✔ Escape Velocity Formula v_e = √(2GM/R)! +4 Marks
✖ Recall v_e = √(2GM/R), so a = 1/2, b = −1/2, c = 1/2.
Concept & Solution:• Equating powers for [L¹ T⁻¹] = [M]ᵃ [L]ᵇ [M⁻¹ L³ T⁻²]ᶜ yields a = 1/2, b = −1/2, c = 1/2.
SI System & Capacitive Reactance
JEE Main 2024 (Apr Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q14. The capacitive reactance is given by X_C = 1 / (ω C), where ω is angular frequency and C is capacitance. What is the dimensional formula of X_C?
✔ Reactance is Resistance! +4 Marks
✖ Remember: Reactance X_C has the exact same unit (Ohms) and dimensions as Resistance R!
Concept & Solution:• [X_C] = 1 / ([ω] [C]) = 1 / ([T⁻¹] × [M⁻¹ L⁻² T⁴ A²]) = [M¹ L² T⁻³ A⁻²].
Astrophysical Dimensional Analysis
JEE Main 2024 (Jan Shift 1)
⏱ JEE Standard Pace: 120s+4 / -1 Scale
Q15. The oscillation period (T) of a gas bubble under adiabatic expansion in a liquid of density ρ, pressure P, and radius R depends as T ∝ Pᵃ ρᵇ Rᶜ. The values of (a, b, c) are:
✔ Adiabatic Bubble Pulsation Solved! +4 Marks
✖ Set up equations for M, L, and T: [M⁰ L⁰ T¹] = [M L⁻¹ T⁻²]ᵃ [M L⁻³]ᵇ [L]ᶜ.
Concept & Solution:• Equating powers gives a = −1/2, b = 1/2, c = 1 (T ∝ R √(ρ / P)).
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