Welcome to the study of Conservation of Angular Momentum, a fundamental principle in rotational mechanics crucial for JEE Physics. This concept helps us understand how objects rotate and interact in the absence of external torques. Mastering this topic will equip you to solve a wide range of problems involving spinning objects, from ice skaters to planetary motion.

Angular momentum (L) is the rotational equivalent of linear momentum. For a point mass, it's defined as the cross product of its position vector (r) from the axis of rotation and its linear momentum (p = mv): L = r x p. For a rigid body rotating about a fixed axis, it's given by L = Iω, where I is the moment of inertia and ω is the angular velocity.

This principle has numerous real-world applications and is frequently tested in JEE. Let's explore some key scenarios.

When an ice skater pulls their arms in, their moment of inertia (I) decreases. Since angular momentum (L = Iω) is conserved (assuming negligible friction), their angular velocity (ω) must increase to compensate. This is why they spin faster.

Similarly, a diver or gymnast can control their rotation speed by tucking their body (decreasing I, increasing ω) or extending it (increasing I, decreasing ω). This allows them to perform flips and twists with precision.

The Earth orbiting the Sun is another example. As the Earth moves closer to the Sun in its elliptical orbit, its moment of inertia decreases, and its orbital speed increases. Conversely, when it's farther away, its speed decreases. This is a direct consequence of the conservation of angular momentum.

Remember these core relationships for solving problems:

Angular Momentum (Point Mass): $L = r \times p = m(r \times v)$

Angular Momentum (Rigid Body): $L = I\omega$

Conservation of Angular Momentum: $L_{initial} = L_{final}$

Moment of Inertia (Examples):

Solid Cylinder/Disk (about central axis): $I = \frac{1}{2}MR^2$

Hollow Cylinder/Ring (about central axis): $I = MR^2$

Solid Sphere (about diameter): $I = \frac{2}{5}MR^2$

Hollow Sphere (about diameter): $I = \frac{2}{3}MR^2$

1. Identify the system.
2. Check if the net external torque on the system is zero. If yes, angular momentum is conserved.
3. Define the initial and final states of the system.
4. Calculate the initial angular momentum (L_initial = Σ I_i ω_i).
5. Calculate the final angular momentum (L_final = Σ I'_i ω'_i).
6. Equate L_initial and L_final and solve for the unknown variable.

Work through practice problems involving rotating discs, spheres, and systems where mass distribution changes. Pay close attention to how the moment of inertia is calculated for different shapes and configurations.