Introduction: Angular momentum (often denoted L) measures the “quantity of rotation” of an object. It is the rotational analogue of linear momentum. For a point mass, L = r × p, where r is the position vector from the rotation axis (or origin) to the mass and p = m v is its linear momentum. In fixed-axis rotation of a rigid body, all particles rotate about the same line, and their angular momenta add up to a total L = I ω, directed along the axis. Here I is the moment of inertia (rotational inertia) and ω is the angular speed. Angular momentum is a vector (direction given by the right-hand rule) and, like linear momentum, it is conserved when no external torque acts on the system. For example, a spinning ice skater (or a person on a swivel chair) speeds up by pulling in her arms: her moment of inertia I decreases and her angular speed ω increases so that Iω remains constant. This conservation principle explains many phenomena from the stability of spinning gyroscopes (shown below) to planetary orbits and ice skating tricks.



Definitions and Key Concepts:

Angular Momentum (particle): L = r × p, the cross-product of position r and linear momentum p. Its magnitude is L = m v⊥ r, where r is the lever arm perpendicular to velocity.

Rigid Body, Fixed Axis: In fixed-axis rotation, each particle in the body moves in a circle about the axis. The total angular momentum about that axis is L = I ω (vector along the axis).

Moment of Inertia (I): Defined by I = Σ m_i r_i² (sum over all mass elements at distance r_i from the axis). It quantifies the mass distribution: more mass or larger radius ⇒ larger I, more “rotational inertia.”

Torque (τ): The rotational analogue of force. It is defined as the rate of change of angular momentum: Στ = dL/dt. If the net external torque is zero, dL/dt = 0 and the total angular momentum is conserved.

Rotational Analogies: Many linear-motion formulas have rotational counterparts: force F ↔ torque τ = Iα, momentum p = m v ↔ L = Iω, and kinetic energy K = ½m v² ↔ K = ½Iω². (See below for formulas.)

Derivations and Formulas:

A rigid body’s angular momentum about its fixed axis can be derived by summing particle contributions. Each particle of mass m at distance r moving at speed v = ωr contributes l = mωr² about the axis. Summing gives L = (Σ m_i r_i²) ω = I ω, confirming L = I ω along the axis.

Rotational Kinetic Energy: K = ½ I ω². (Analogous to linear ½mv².)

L = I ω: (for symmetric bodies about the axis). If the body is symmetric, all angular momentum lies on the axis so L = Iω.

Centrifugal Force (context): When dealing with rotation, remember v = ωr, so linear and rotational formulas mix via r.

Key Formulas (rotational vs. linear):

- Torque: F = m a ↔ τ = I α

- Momentum: p = m v ↔ L = I ω (for fixed axis)

- Kinetic Energy: K = ½ m v² ↔ K = ½ I ω².

- Work/Energy: Work = F·s ↔ Work = τ·θ.

- Impulse: Δp = F Δt ↔ ΔL = τ Δt.

Moment of Inertia and Its Role:

The moment of inertia I plays the role of “mass” in rotation. It depends on total mass and how far that mass is from the axis. A large I (mass concentrated far out) means the body strongly resists changes in its spin. For example, a heavy flywheel (large I) spins stably even if bumped. For common shapes rotating about symmetry axes, standard formulas are: thin hoop I = M R², solid disc I = (½)MR², solid sphere I = (2/5)MR², uniform rod about center I = (1/12)M L². These are derived by integrating Σ m r² for the shape. In equations, I appears in L = Iω, in rotational KE, and in τ = Iα. A change in I leads to an inverse change in ω if L is conserved (see below).

Conservation of Angular Momentum:

When the net external torque on a system is zero, its total angular momentum remains constant. In a fixed-axis scenario, this means I₁ω₁ = I₂ω₂ before and after any internal changes. For example, if a rotating person pulls arms inward (reducing I), her angular speed ω increases so that Iω stays the same. NCERT demonstrates this with a student on a swivel chair: extending arms slows the spin, pulling them in speeds it up. Similarly, a spinning ice skater can control her spin rate by changing body shape. Astrophysically, conservation explains why collapsing stars (neutron stars) spin extremely fast: their radius (and thus I) suddenly drops, so ω must rise to keep L constant.

Read Also: Mechanical Properties of Solids: Stress and Strain VAVA Classes

Examples and Problem-Solving Strategies:

Identify Known Quantities: Note whether you have angular speed ω, moment of inertia I, torque τ, or kinetic energy K. For a rigid body, first compute I (using shape formulas if needed) or use given L.

Use the Right Formula: If calculating L directly: use L = I ω. If given linear values for a point mass at distance r from axis: use L = m v r or L = m v⊥ r. For kinetic energy problems, use K = ½ I ω². If rotational work or torque is involved, use Στ = dL/dt.

Conservation (I changes): If an internal change (no external torque) alters I, set I₁ ω₁ = I₂ ω₂. Then solve for the unknown ω or I. For example, a common physics problem asks: “A flywheel has KE = 360 J at ω = 30 rad/s. Find its moment of inertia.” Here use K = ½ I ω² ⇒ I = 2K/ω². Plugging in gives I = 2(360)/(30²) = 0.80 kg·m². (Vedantu’s worked solution follows this approach.)

Torque Applications: If a known torque acts for time Δt, find change in L by ΔL = τ Δt. Then new ω can be found from L = I ω.

Use Analogies: Treat angular problems like linear ones: e.g. equate “rotational momentum” instead of linear. Helpful analogies: L like p = m v, τ like F, etc. Draw free-body or free-body-angular diagrams if needed.

Conceptual Analogies and Visuals:

A spinning bicycle wheel (or gyroscope) resists tipping – the wheel’s angular momentum keeps its axis pointing steady.

A merry-go-round: if no one pushes (no external torque), its spin stays constant. Pushing on the edge is like applying τ to change L.

Imagine water swirling: pulling arms in (reducing radius) speeds up just like a figure skater.

Conclusion: Angular momentum about a fixed axis is given by L = Iω and is the rotational counterpart of linear momentum. The moment of inertia I captures how mass is distributed relative to the axis. A key result is that if no external torque acts, Iω stays constant. This conservation law explains why changing a body’s shape changes its rotation rate. In problem-solving, identify which formula applies (e.g. L = Iω, K = ½Iω², or I₁ω₁=I₂ω₂) and apply it step-by-step. These notes have summarized the definitions, formulas, and strategies needed to master angular momentum in Class 11 rotational motion.