Class 11 > Unit # 04: Rotational & Circular Motion > Law of Conservation of Angular Momentum

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Unit No. 4 Rotational and Circular Motion - Angular Momentum & Conservation Law

Define Angular Momentum. State and Prove Law of Conservation of Angular Momentum.

Angular Momentum

"Angular momentum is the rotational analogue of linear momentum, defined as the product of the moment of inertia and angular velocity, or mathematically as the cross product of position vector and linear momentum vector."

It represents the quantity of rotation of a body and resists changes in its rotational state.

Units and Dimensional Formula

• SI Unit: Kilogram square meter per second ($\text{kg}\cdot\text{m}^2\cdot\text{s}^{-1}$)
• Dimensional Formula: $[\text{M}][\text{L}]^2[\text{T}]^{-1}$

Mathematical Expression

For a point particle of mass $m$ at position $\vec{r}$ with linear momentum $\vec{p}$, the angular momentum $\vec{L}$ is:

$\vec{L} = \vec{r} \times \vec{p}$

The magnitude of angular momentum is given by:

$L = r p \sin\theta = r (mv) \sin\theta$

For circular motion where the position vector is perpendicular to linear momentum ($\theta = 90^\circ$):

$L = mvr$

Since linear velocity $v = r\omega$, substituting this into the equation gives:

$L = (mr^2)\omega = I\omega$

Law of Conservation of Angular Momentum

Statement

"When no net external torque acts on a system, the total angular momentum of the system remains constant."

Mathematical Proof

According to Newton's second law for rotation, torque ($\tau$) is defined as the rate of change of angular momentum with respect to time:

$\tau = \frac{d\vec{L}}{dt}$

If the net external torque acting on the body or system is zero ($\tau_{\text{net}} = 0$):

$\frac{d\vec{L}}{dt} = 0$

Integrating this equation with respect to time implies that the angular momentum vector is constant:

$\vec{L} = \text{constant} \quad \implies \quad I_1\omega_1 = I_2\omega_2$

Conclusion:
This relation proves that if the moment of inertia $I$ decreases (e.g., when a spinning figure skater pulls their arms inward), the angular velocity $\omega$ must increase proportionally to conserve total angular momentum. Conversely, if $I$ increases, $\omega$ decreases.

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