Formula Bank System of Particles & Rotational Motion

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🧠 How to Use This Page

Don't memorise formulas in isolation. For every formula, understand: (1) What physical quantity it links. (2) What happens at limits (θ=0, ω=0, etc.). (3) What the examiner tests. Use the dimensional analysis column to verify your answers.

📍 Centre of Mass

COM of Discrete System Core

R_cm = Σmᵢrᵢ / M

Dim: [L] All Exams

M = total mass. Works in x, y, z separately. Foundation of the chapter.

COM — Continuous Body JEE Level

R_cm = ∫r dm / M

Dim: [L] JEE Adv

Use linear, surface, or volume density as appropriate. Key for derivation questions.

Velocity of COM Core

v_cm = Σmᵢvᵢ / M = p_total/M

Dim: [LT⁻¹] All Exams

If external force = 0, v_cm = constant. Newton's 1st law for system.

Acceleration of COM Core

a_cm = F_ext/M = ΣF_ext/M

Dim: [LT⁻²] All Exams

Internal forces cancel out. Only external forces determine COM acceleration.

COM — Semicircular Ring Exam Trap

y_cm = 2R/π

Dim: [L] NEET/JEE

≈ 0.637R. Measured from the diameter. Derived via y_cm = ∫y dℓ / (πR).

COM — Semicircular Disc Exam Trap

y_cm = 4R/3π

Dim: [L] NEET/JEE

≈ 0.424R. Less than ring because disc has more mass near centre.

🔄 Rotational Kinematics

Kinematic Equations (Rotation) Core

ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ
θ = (ω + ω₀)/2 × t

All Exams

Exact analogy with linear kinematics. Replace s→θ, v→ω, u→ω₀, a→α.

Angular & Linear Relation Core

s = rθ · v = rω · aₜ = rα

All Exams

r = distance from axis. Valid for any point on rigid body.

Centripetal Acceleration Core

aₙ = rω² = v²/r

Dim: [LT⁻²] All Exams

Directed toward centre. Even in uniform circular motion this is non-zero!

Angular Velocity Core

ω = dθ/dt = 2πf = 2π/T

Dim: [T⁻¹] All Exams

Unit: rad/s. Vector along axis (right-hand rule).

🌀 Torque & Angular Momentum

Torque Core

τ = r × F → |τ| = rF sinθ

Dim: [ML²T⁻²] All Exams

Unit: N·m. θ = angle between r and F. Max torque when θ = 90°.

Newton's 2nd Law (Rotation) Core

τ_net = Iα (also τ = dL/dt)

Dim: [ML²T⁻²] All Exams

τ = dL/dt is the general form. τ = Iα only when I is constant.

Angular Momentum (particle) Core

L = r × p = mvr sinθ

Dim: [ML²T⁻¹] All Exams

For a particle. For particle in straight line: L = mvd (d = perp. distance).

Angular Momentum (rigid body) Core

L = Iω

Dim: [ML²T⁻¹] All Exams

For rotation about fixed axis. Unit: kg·m²/s or J·s.

Conservation of Angular Momentum High Yield

L = Iω = constant (if τ_ext = 0)

Dim: [ML²T⁻¹] All Exams

I₁ω₁ = I₂ω₂. Used in: skater problem, bullet-disc, planet orbit.

Work done by Torque Core

W = τ · θ (constant τ)

Dim: [ML²T⁻²] NEET/JEE

Analogue of W = F·s. Variable τ: W = ∫τ dθ.

⚖️ Moment of Inertia

Definition Core

I = Σmᵢrᵢ² = ∫r² dm

Dim: [ML²] All Exams

Unit: kg·m². Depends on axis AND mass distribution.

Parallel Axis Theorem High Yield

I = I_cm + Md²

Dim: [ML²] All Exams

d = distance from COM axis to new axis. I_cm is minimum. New axis need not pass through body.

Perpendicular Axis Theorem High Yield

I_z = I_x + I_y

Dim: [ML²] Planar Only

ONLY for 2D (planar) bodies. z ⊥ plane of body. x and y are in the plane.

Radius of Gyration Core

I = Mk² → k = √(I/M)

Dim: [L] NEET/JEE

k = effective distance of mass from axis. Used in rolling motion formula.

🎳 Rolling Motion

Condition for Pure Rolling High Yield

v_cm = Rω

All Exams

Contact point has zero velocity. No slipping occurs. Differentiate: a_cm = Rα.

Total KE (Rolling) High Yield

KE = ½mv_cm²(1 + k²/R²)

Dim: [ML²T⁻²] All Exams

= ½mv² + ½Iω². The ratio KE_rot/KE_total = k²/(R²+k²).

Acceleration on Incline High Yield

a = g sinθ / (1 + k²/R²)

Dim: [LT⁻²] All Exams

For pure rolling on incline. Smaller k²/R² → greater acceleration → reaches bottom first.

Velocity at Bottom of Incline Core

v = √[2gh/(1+k²/R²)]

Dim: [LT⁻¹] All Exams

Derived from energy conservation. h = vertical height. Independent of mass and radius.

📏 Dimensional Analysis

Quantity	Symbol	SI Unit	Dimensions	Notes
Angular Displacement	θ	radian (rad)	Dimensionless	rad = m/m
Angular Velocity	ω	rad/s	[T⁻¹]	ω = 2πf
Angular Acceleration	α	rad/s²	[T⁻²]	α = dω/dt
Torque	τ	N·m	[ML²T⁻²]	Same as energy — but NOT energy!
Moment of Inertia	I	kg·m²	[ML²]	—
Angular Momentum	L	kg·m²/s	[ML²T⁻¹]	Same as Planck's constant h
Radius of Gyration	k	m	[L]	k = √(I/M)
Power (rotational)	P	W	[ML²T⁻³]	P = τω

❌ Critical Dimensional Trap

Torque and Energy have the same dimensions [ML²T⁻²] and same SI unit (N·m = J). But they are NOT the same physical quantity. Torque is a vector (cross product), energy is a scalar. In MCQs, this distinction is tested. A body can have energy without torque and vice versa.

Complete Formula Reference

📍 Centre of Mass

🔄 Rotational Kinematics

🌀 Torque & Angular Momentum

⚖️ Moment of Inertia

🎳 Rolling Motion

📏 Dimensional Analysis

🧮 Moment of Inertia Calculator