Class 11 · Chapter 7

System of Particles & Rotational Motion

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System of Particles & Rotational Motion Formula Sheet

13 formulas across 4 topics in System of Particles & Rotational Motion.

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Updated 2026-07-03 · v1.0.1

Moment of Inertia

Moment of inertia is rotational mass — it measures how far the mass sits from the axis, not just how much mass there is.

I = Σ mᵢrᵢ² = ∫ r² dm

Moment of inertia (definition)

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
I	moment of inertia about the chosen axis	kg·m²	[ML²]
rᵢ	perpendicular distance of mass mᵢ from the axis	m	[L]

Valid when

Depends on the AXIS — the same body has different I about different axes

Common mistakes

r is the perpendicular distance from the AXIS, not from any point

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I = I(cm) + Md²

Parallel axis theorem

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
I(cm)	moment of inertia about the parallel axis through the COM	kg·m²	[ML²]
d	distance between the two parallel axes	m	[L]
M	total mass	kg	[M]

Valid when

The reference axis MUST pass through the centre of mass

Common mistakes

Applying it between two arbitrary parallel axes — one of them must be the COM axis; go via the COM in two steps otherwise

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I(z) = I(x) + I(y)

Perpendicular axis theorem

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
I(z)	MOI about the axis perpendicular to the lamina	kg·m²	[ML²]
I(x), I(y)	MOI about two perpendicular in-plane axes meeting the z-axis	kg·m²	[ML²]

Valid when

PLANAR bodies (laminas) only — invalid for spheres, cylinders, cubes

Common mistakes

Using it on a 3-D body is the most common theorem misuse in this chapter

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Worth remembering

Standard MOIs (about symmetry axis through COM): ring MR², disc ½MR², solid sphere ⅖MR², hollow sphere ⅔MR², rod (centre) ML²/12, rod (end) ML²/3, solid cylinder ½MR²
Radius of gyration: k = √(I/M) — the distance at which all mass could sit and give the same I

NEET asks the standard values directly; JEE combines both theorems on composite bodies — always route through the COM axis.

Torque & Rotational Dynamics

Torque is to rotation what force is to translation. The whole of rotational dynamics is Newton's second law with I replacing m and α replacing a.

τ = r F sinθ

Torque of a force

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
τ	torque about the chosen point/axis	N·m	[ML²T⁻²]
r	position vector magnitude from axis to point of application	m	[L]
θ	angle between r and F	rad (or °)	[M⁰L⁰T⁰]

Valid when

r sinθ is the lever arm — the perpendicular distance from the axis to the LINE of the force

Common mistakes

A force whose line of action passes through the axis has ZERO torque, however large the force

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τ(net) = Iα

Rotational form of Newton's second law

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
α	angular acceleration	rad/s²	[T⁻²]
I	moment of inertia about the rotation axis	kg·m²	[ML²]

Valid when

About a fixed axis, or about the COM axis even when the COM accelerates

Common mistakes

Pulley problems with massive pulleys: the two string tensions are DIFFERENT — their difference provides the pulley's τ = Iα

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K(rot) = ½Iω², W = τθ, P = τω

Rotational kinetic energy, work and power

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
ω	angular speed	rad/s	[T⁻¹]
θ	angle turned	rad	[M⁰L⁰T⁰]

Valid when

Perfect analogues of ½mv², W = Fs and P = Fv with m→I, v→ω, F→τ, s→θ

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Worth remembering

Every translational equation has a rotational twin — build the analogy table once and the chapter halves in size

Angular Momentum

Angular momentum is conserved whenever net external torque vanishes — the physics behind spinning skaters, shrinking orbits, and neutron stars.

L = Iω (rigid body), L = mvr sinθ (particle)

Angular momentum

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
L	angular momentum about the axis/point	kg·m²/s	[ML²T⁻¹]
r sinθ	perpendicular distance from the point to the velocity line	m	[L]

Valid when

τ = dL/dt — the rotational twin of F = dp/dt

Common mistakes

A particle moving in a STRAIGHT line still has angular momentum about any point off that line: L = mv × (perpendicular distance)

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I₁ω₁ = I₂ω₂ (when τ(ext) = 0)

Conservation of angular momentum

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τ(ext) = dL/dt = 0 ⟹ L constant

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
I₁, I₂	moments of inertia before and after	kg·m²	[ML²]
ω₁, ω₂	angular speeds before and after	rad/s	[T⁻¹]

Valid when

Net external TORQUE zero (forces may exist — a skater's weight and normal force have no torque about the spin axis)

Common mistakes

When a skater pulls arms in, KE INCREASES (K = L²/2I with L fixed, I smaller) — the extra energy comes from muscular work

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∫τ dt = ΔL

Angular impulse–angular momentum theorem

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Integrating τ = dL/dt over the interaction time

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
∫τ dt	angular impulse about the axis	N·m·s	[ML²T⁻¹]
ΔL	change in angular momentum	kg·m²/s	[ML²T⁻¹]

Valid when

The rotational twin of J = Δp — use it for sudden hits, sticky collisions with rods, and struck billiard balls

Common mistakes

A rod hit at its centre of percussion produces zero reaction at the pivot — the sweet spot of a cricket bat

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Worth remembering

Kepler's second law (equal areas) is just conservation of angular momentum in disguise: dA/dt = L/2m

Rolling Motion

Rolling without slipping ties translation to rotation through v = ωR — one constraint that unlocks every rolling problem.

v(cm) = ωR, a(cm) = αR

Rolling without slipping condition

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v(cm)	speed of the centre	m/s	[LT⁻¹]
R	radius of the rolling body	m	[L]

Valid when

Contact point is instantaneously at REST relative to the ground

Common mistakes

Speed of the TOP point is 2v(cm), bottom point is 0 — a favourite one-mark trap

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K = ½mv² (1 + k²/R²)

Total KE of a rolling body

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K(trans) + K(rot) with ω = v/R and I = mk²

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
k	radius of gyrationk²/R²: ring 1, disc ½, solid sphere ⅖, hollow sphere ⅔	m	[L]

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a = g sinθ / (1 + k²/R²)

Acceleration rolling down an incline

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Energy method or force-torque pair on the incline

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
θ	incline angle	rad (or °)	[M⁰L⁰T⁰]
k²/R²	shape factor	dimensionless	[M⁰L⁰T⁰]

Valid when

Rolling without slipping (needs enough friction: it supplies torque but does NO work here)
Independent of mass and radius — only the SHAPE matters

Common mistakes

Race down an incline: solid sphere beats disc beats hollow sphere beats ring — smaller k²/R² wins

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L(contact) conserved: mv₀R = mvR + Iω ⟹ v = v₀/(1 + k²/R²)

Slipping → rolling transition (angular momentum about the contact point)

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Friction acts AT the contact point, so its torque about that point is zero — L about the contact line is conserved

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v₀	initial sliding speed (no spin)	m/s	[LT⁻¹]
v	final speed when pure rolling begins (ω = v/R)	m/s	[LT⁻¹]
k	radius of gyrationsolid sphere: v = 5v₀/7; disc: v = 2v₀/3; ring: v = v₀/2	m	[L]

Valid when

Horizontal surface, no external torque about the contact line other than friction (which passes through it)

Common mistakes

Kinetic energy is NOT conserved during the skid — friction burns some; only L about the contact point survives

Read the concept →

Worth remembering

Friction in perfect rolling on a FIXED incline does zero work — mechanical energy is conserved even though friction acts

NEET tests the incline race and KE split; JEE Advanced adds slipping-to-rolling transitions where angular momentum about the contact point is the shortcut.

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