Conservation of Angular Momentum — Revision Notes

⚡ 30-Second Revision

Condition: — Net external torque $\vec{\tau}_{ext} = 0$.
Principle: — Total angular momentum $\vec{L}$ of the system remains constant.
Formula (rigid body): — $L = I\omega = \text{constant}$.
Conservation Equation: — $I_1\omega_1 = I_2\omega_2$.
Angular Momentum (particle): — $\vec{L} = \vec{r} \times \vec{p}$.
Torque: — $\vec{\tau} = \vec{r} \times \vec{F}$.
Rotational Kinetic Energy: — $K_{rot} = \frac{1}{2}I\omega^2 = \frac{L^2}{2I}$. (NOT conserved if $I$ changes, even if $L$ is conserved).
Key Examples: — Ice skater, diver, planetary motion, rotating platforms, rotational collisions.

2-Minute Revision

The Conservation of Angular Momentum is a cornerstone of rotational dynamics. It states that the total angular momentum ($\vec{L}$) of a system remains constant if the net external torque ($\vec{\tau}_{ext}$) acting on it is zero.

This is a direct consequence of Newton's second law for rotation, $\vec{\tau}_{ext} = d\vec{L}/dt$. For a rigid body, angular momentum is given by $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.

Therefore, if $I$ changes (e.g., by mass redistribution), $\omega$ must change inversely to keep $I\omega$ constant ($I_1\omega_1 = I_2\omega_2$). Common applications include ice skaters pulling in their arms to spin faster, divers tucking their bodies for somersaults, and the constant areal velocity of planets in elliptical orbits (Kepler's second law).

Remember that angular momentum is a vector, and its conservation implies both constant magnitude and direction. Crucially, rotational kinetic energy ($K_{rot} = \frac{1}{2}I\omega^2$) is generally *not* conserved when angular momentum is conserved if the moment of inertia changes; internal work is done to change $I$, altering $K_{rot}$.

5-Minute Revision

The principle of Conservation of Angular Momentum is fundamental: if the net external torque on a system is zero, its total angular momentum remains constant. This means $\vec{L}_{initial} = \vec{L}_{final}$.

For a rigid body rotating about a fixed axis, this translates to $I_1\omega_1 = I_2\omega_2$. Here, $I$ is the moment of inertia (rotational inertia) and $\omega$ is the angular velocity. The moment of inertia depends on the mass distribution; pulling mass closer to the axis decreases $I$, while moving it further increases $I$.

Example 1: Ice Skater

An ice skater spinning with arms outstretched has an initial moment of inertia $I_1$ and angular velocity $\omega_1$. When she pulls her arms in, her moment of inertia decreases to $I_2 < I_1$. To conserve angular momentum, her angular velocity must increase: $\omega_2 = (I_1/I_2)\omega_1$. Since $I_1/I_2 > 1$, $\omega_2 > \omega_1$. The work done by her muscles to pull her arms in increases her rotational kinetic energy ($K_{rot} = L^2/(2I)$), as $I$ decreases while $L$ is constant.

Example 2: Rotating Platform

A person standing on a rotating platform drops a mass onto it. The initial angular momentum is $L_1 = I_{platform}\omega_1$. After the mass $m$ is dropped at a distance $r$ from the axis, the new moment of inertia is $I_2 = I_{platform} + mr^2$. The new angular velocity $\omega_2$ is found from $I_1\omega_1 = I_2\omega_2$, so $\omega_2 = \frac{I_{platform}\omega_1}{I_{platform} + mr^2}$. Since $I_2 > I_1$, $\omega_2 < \omega_1$.

Key Points for NEET:

1
Condition: — Always check for zero net external torque. Internal forces/torques do not affect total angular momentum.
2
Vector Nature: — Angular momentum is a vector. Conservation implies both magnitude and direction are constant.
3
Moment of Inertia: — Be proficient in calculating $I$ for various shapes and configurations (point masses, discs, rods, spheres) and using parallel/perpendicular axis theorems.
4
Kinetic Energy: — Remember that rotational kinetic energy is generally *not* conserved if $I$ changes, even if $L$ is. The change in $K_{rot}$ is due to internal work done.
5
Applications: — Be familiar with common examples like planetary motion (Kepler's 2nd law) and rotational collisions.

Prelims Revision Notes

Conservation of Angular Momentum (CoAM) is a critical concept in rotational dynamics for NEET. It states that the total angular momentum ($\vec{L}$) of a system remains constant if the net external torque ($\vec{\tau}_{ext}$) acting on it is zero. This is a direct consequence of Newton's second law for rotation: $\vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$. If $\vec{\tau}_{ext} = 0$, then $\vec{L} = \text{constant}$.

Formulas to Remember:

Angular momentum for a rigid body: $L = I\omega$
Angular momentum for a particle: $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v})$
Conservation equation: $I_1\omega_1 = I_2\omega_2$
Rotational kinetic energy: $K_{rot} = \frac{1}{2}I\omega^2 = \frac{L^2}{2I}$
Moment of inertia for common shapes: Disc ($I = \frac{1}{2}MR^2$), Rod about center ($I = \frac{1}{12}ML^2$), Rod about end ($I = \frac{1}{3}ML^2$), Solid sphere ($I = \frac{2}{5}MR^2$), Hollow sphere ($I = \frac{2}{3}MR^2$).
Parallel axis theorem: $I = I_{CM} + Md^2$
Perpendicular axis theorem (for planar bodies): $I_z = I_x + I_y$

Key Concepts for Recall:

Condition: — Zero net external torque. Internal torques cancel out.
Vector Nature: — $\vec{L}$ is a vector; its direction is also conserved.
Mass Redistribution: — Changes in mass distribution affect $I$, thus affecting $\omega$ to conserve $L$.
Kinetic Energy vs. Angular Momentum: — $K_{rot}$ is generally NOT conserved when $L$ is conserved if $I$ changes. The work done by internal forces accounts for the change in $K_{rot}$.
Applications: — Ice skaters, divers, planetary motion (Kepler's 2nd Law), rotating platforms, rotational collisions (e.g., bullet hitting a rod). Always identify the system and the axis of rotation correctly. For collisions, choose the pivot where external forces (like pivot reaction) don't exert torque.

Vyyuha Quick Recall

Let It Often Conserve: L (Angular Momentum) is conserved if Internal forces only act, or Outside torques Cancel out.