Q.Define angular momentum, and prove that the rate of change of angular momentum equals the external torque. OR Show that the area of the triangle formed by vectors a and b is one-half of the magnitude of a x b.
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🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Angular Momentum
Angular Momentum: The Rotational Cousin of Momentum
You already know linear momentum — how hard it is to stop a moving object. A truck moving at 20 m/s has more momentum than a bicycle at the same speed. Now imagine something spinning: a bicycle wheel, a spinning top, or a figure skater pulling their arms in. There is a similar "quantity of motion" for rotation, and that is angular momentum.
The core intuition is simple: angular momentum measures how much rotation an object has, and how hard it is to change that rotation. Just as a heavy truck is hard to stop, a heavy flywheel spinning fast is hard to stop spinning.
The Two Faces of Angular Momentum
Angular momentum appears in two forms, depending on what you are studying.
For a single particle moving in a straight line or a curve, angular momentum is defined relative to a chosen point (usually the centre of rotation). It is the cross product of the position vector and the linear momentum:
L=r×p
Here r is the vector from the reference point to the particle, and p=mv is its linear momentum. The magnitude is L=rpsinθ, where θ is the angle between r and p. This tells you: the farther the particle is from the point, and the faster it moves perpendicular to that line, the greater its angular momentum.
For a rigid body rotating about a fixed axis, the formula simplifies beautifully. Every particle in the body contributes, and when you add them all up, you get:
L=Iω
where I is the moment of inertia (the rotational analogue of mass) and ω is the angular velocity (how fast it spins). This is the direct parallel of p=mv.
Linear: p=mv⟷Rotational: L=Iω
Why Angular Momentum Matters
The real power of angular momentum is its conservation. In the absence of an external torque (the rotational analogue of force), angular momentum stays constant. This is why a figure skater spins faster when she pulls her arms in — her moment of inertia I decreases, so ω must increase to keep L constant.
Conservation of Angular Momentum: If net external torque τext=0, then L is constant in both magnitude and direction.
This principle explains everything from why a bicycle stays upright to why neutron stars spin at incredible speeds after a supernova collapse.
Connecting the Two Definitions
The particle definition L=r×p is the fundamental one. The rigid-body formula L=Iω is derived from it by summing over all particles in the body. For a single particle moving in a circle of radius r with speed v, you get L=rmv=mr2ω=Iω, since I=mr2 for that particle.
So the two definitions are not separate — they are the same idea at different levels of description.
A Quick Check on Direction …
Angular momentum measures the quantity of rotational motion a body possesses about a chosen axis or point, and its rate of change equals the net torque acting on the body. …
Angular momentum L = r x p. Differentiating with respect to time shows dL/dt equals the net external torque tau = r x F.
Definition: The angular momentum of a particle about a point O is defined as the cross (vector) product of its position vector r (from O) and its linear momentum p = mv:
L = r x p
Proof that dL/dt = tau:
Differentiating L = r x p with respect to time:
dL/dt = d(r x p)/dt = (dr/dt) x p + r x (dp/dt)
Now, dr/dt = v (velocity), and p = mv, so:
(dr/dt) x p = v x (mv) = m(v x v) = 0 (since the cross product of any vector with itself is zero).
Also, dp/dt = F (the net force acting on the particle, by Newton's second law).
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Showing the 12 most recent of 20 on this concept.
- CBSE 2026Set ANNUAL1 markQ.Write S. I. unit of Angular Momentum.
›Reveal solutionSolution
L=Iω has SI unit kgm2s−1, same as J·s.
Angular momentum is L=r×p=Iω, where moment of inertia I has SI unit kgm2 and angular velocity ω has SI unit s−1 (rad/s, but radian is dimensionless). Multiplying:
[L]=kgm2×s−1=kgm2s−1 …
- CBSE 2026Set ANNUAL1 markMCQQ.A particle is moving with a constant velocity along a straight line parallel to positive x-axis. The magnitude of its angular momentum with respect to origin is:(a) decreasing with x(b) zero(c) remaining constant(d) increasing with x
›Reveal solutionSolution
Angular momentum about the origin depends on the perpendicular distance from the origin to the particle's line of motion; for straight-line motion parallel to the x-axis, this perpendicular distance (the particle's fixed y-coordinate) never changes, so L is constant.
The angular momentum of a particle about the origin is
L = r x p = m(r x v)
For a particle moving along a straight line parallel to the positive x-axis at a fixed height y = y0, with constant velocity v = vx-hat, the position vector is r = xx-hat + y0*y-hat.
L = m(r x v) = m[(xx-hat + y0y-hat) x (vx-hat)] = m[xv*(x-hat x x-hat) + y0v(y-hat x x-hat)] = -mvy0*z-hat
…
- CBSE 2026Set ANNUAL1 markQ.Fill in the blank: Moment of linear momentum is called ____.
›Reveal solutionSolution
The moment of linear momentum about a point is called angular momentum.
Just as the moment of a force about a point is called torque (τ = r × F), the moment of linear momentum p about a point is defined as angular momentum:
L = r × p …
- CBSE 2025Set ANNUAL1 markMCQQ.The correct relation between radius of gyration (k) and moment of inertia (I) is(a) (A) k = √(m/I)(b) (B) k = √(I/m)(c) (C) k = Im²(d) (D) k = ml²
›Reveal solutionSolution
[!TLDR]
(B) k = √(I/m)
Why
By definition I = mk², so k = √(I/m …
- CBSE 2024Set ANNUAL1 markMCQQ.A particle is moving with constant velocity parallel to x-axis. Its angular momentum relative to origin point (A) is zero (B) remains constant (C) goes on increasing (D) goes on decreasing
›Reveal solutionSolution
A particle in straight-line uniform motion has constant angular momentum about any fixed point (unless it passes through that point along the line).
Angular momentum about the origin is L=r×p, with magnitude L=p⋅d, where d is the perpendicular distance from the origin to the line along which the particle moves. Since the particle moves with constant velocity, p=mv is constant in both m …
- CBSE 2024Set ANNUAL1 markMCQQ.A body of mass M is moving with uniform angular velocity ω about its axis of rotation. I is its moment of inertia about this axis. Its angular momentum will be (A) ½Iω^2 (B) MIω^2 (C) I^2ω (D) Iω
›Reveal solutionSolution
Angular momentum of a rotating body is L=Iω.
Just as linear momentum is p=mv, angular momentum about the rotation axis is defined as L=Iω, where I is the moment of inertia about that axis and ω is the a …
- CBSE 2024Set ANNUAL1 markMCQQ.The product of moment of inertia and angular velocity is called(a) Torque(b) Impulse(c) Linear momentum(d) Angular momentum
›Reveal solutionSolution
Angular momentum L is defined as the product of moment of inertia I and angular velocity ω: L = Iω.
This is the rotational analogue of linear momentum p = mv, with mass replaced by moment of inertia and linear velocity replaced by angular velocity. Torque is the rotational analogue of force, impulse is force …
- CBSE 2024Set sz1 markMCQQ.Angular momentum is a: (A) Polar vector (B) Axial vector (C) Scalar (D) None of these
›Reveal solutionSolution
Angular momentum is an axial (pseudo) vector because it is defined as the cross product of two polar vectors, position and linear momentum.
Angular momentum is defined as L=r×p.
Both r (position) and p (linear momentum) are polar (true) vectors — their direction is along an actual physical displacement or motion. The cross product of two polar vectors, however, produces an axial vector (also called a pseudovector): its direction is assigned by convention (the right-hand rule) perpendicular to the plane cont …
- CBSE 2023Set ANNUAL1 markMCQQ.A rigid body rotates with an angular momentum L. If its kinetic energy is halved, the angular momentum becomes :(a) 2L(b) L(c) L/√2(d) L/2
›Reveal solutionSolution
Since KE = L^2/(2I), halving KE while keeping I fixed means L is scaled by 1/sqrt(2).
The rotational kinetic energy of a rigid body is
KE = (1/2) I omega^2
Angular momentum is L = I omega, so omega = L/I. Substituting:
KE = (1/2) I (L/I)^2 = L^2 / (2I)
So L = sqrt(2 x I x KE), i.e. L is proportional to sqrt(KE) for a fixed I (no external torque changes I here).
…
- CBSE 2023Set ANNUAL1 markMCQQ.SI unit of angular momentum is:(a) Joule x Second(b) Newton x Meter(c) kg x m^2(d) Newton x m / Sec
›Reveal solutionSolution
Angular momentum is measured in Joule-second (equivalently kg m^2/s), NOT in kg x m^2 alone or in Newton x metre.
Angular momentum L = I*omega, where moment of inertia I has units kg m^2 and angular velocity omega has units rad/s (dimensionless rad, so effectively 1/s). Hence:
[L] = kg m^2 x (1/s) = kg m^2/s
Now check the given options:
- Joule x Second = (kg m^2/s^2) x s = kg m^2/s -- matches
- Newton x Metre = (kg m/s^2) x m = kg m^2/s^2 -- this is the unit of TORQUE, not angular momentum …
- CBSE 2023Set ANNUAL1 markMCQQ.Match the column: Angular momentum L — match with the correct expression.(a) sqrt(2gR)(b) sqrt(T/m)(c) GMm/r^2(d) I*omega(e) 2pisqrt(l/g)(f) sqrt(gR)(g) m*R^2
›Reveal solutionSolution
Angular momentum is L = Iomega, the rotational counterpart of linear momentum p = mv, matching option (d).
Just as linear momentum p = mv combines mass (inertia for straight-line motion) with linear velocity, angular momentum L combines moment of inertia I (inertia for rotational motion) with angular velocity omega: L = Iomega
…
- CBSE 2022Set TERM11 markMCQQ.Moment of linear momentum is(1) Couple(2) Torque(3) Impulse(4) Angular momentum
›Reveal solutionSolution
'Moment of linear momentum' is the literal definition of angular momentum, L = r x p (position vector crossed with linear momentum), just as torque is defined as the moment of force.
In rotational mechanics, taking the 'moment' of a vector quantity about a point means crossing the position vector r (from that point to where the quantity acts) with the quantity itself.
- Moment of FORCE = r x F = Torque
- Moment of LINEAR MOMENTUM = r x p = Angular Momentum, denoted L …
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