6.12 Angular Momentum in Case of Rotations about a Fixed Axis – Class 11 Physics

After developing rotational kinematics and dynamics, we now arrive at one of the most powerful and elegant quantities in mechanics – angular momentum. In earlier sections, we established the equation and studied rotational kinetic energy. Now we focus specifically on angular momentum for rigid bodies rotating about a fixed axis.

Angular momentum is not merely another formula in rotational mechanics. It is a conserved quantity that simplifies many complex rotational problems. In fact, several advanced JEE Main and JEE Advanced problems become significantly easier when solved using conservation of angular momentum rather than direct torque analysis.

In this section, we will:

• Revisit angular momentum of a particle
• Derive angular momentum of a rigid body about a fixed axis
• Establish the torque–angular momentum relation
• Study conservation of angular momentum in depth
• Explore energy implications
• Solve advanced conceptual scenarios

This completes the conceptual framework of rotational mechanics in Chapter 6.

1. Angular Momentum of a Particle (Revision Foundation)

For a particle of mass moving with velocity , angular momentum about a reference point is defined as:

Where:

• is position vector from reference point
• is linear momentum

Magnitude of angular momentum:

If velocity is perpendicular to radius (pure circular motion):

Using :

This expression shows that angular momentum depends on:

• Mass
• Distance from axis
• Angular velocity

This particle result directly leads to rigid body angular momentum.

2. Angular Momentum of a Rigid Body about a Fixed Axis

Consider a rigid body rotating about a fixed axis with angular velocity .

Each particle of the body moves in a circle about the axis.

For the particle:

Total angular momentum of the rigid body is the sum over all particles:

Since is common for all particles in rigid body rotation:

But by definition of moment of inertia:

Therefore:

This is the fundamental expression for angular momentum of a rigid body about a fixed axis.

Important condition:

This formula is valid when:

• Rotation occurs about a fixed axis
• Moment of inertia about that axis is constant

3. Vector Form and Direction

In full vector notation:

The direction of angular momentum is along the axis of rotation.

Using the right-hand rule:

• Curl fingers in direction of rotation
• Thumb gives direction of

For anticlockwise rotation (viewed from above), points upward.

This directional property becomes crucial in three-dimensional rotational problems.

4. Relation Between Torque and Angular Momentum

From fundamental principles of rotational dynamics:

This equation is the rotational analogue of:

For fixed axis rotation with constant :

Thus torque changes angular momentum in the same way that force changes linear momentum.

If:

Then:

Hence:

This leads to conservation of angular momentum.

5. Conservation of Angular Momentum (Deep Analysis)

If net external torque about the axis is zero:

Then angular momentum remains conserved:

For fixed axis rotation:

This equation applies even when moment of inertia changes.

5.1 Ice Skater Example

When a skater pulls arms inward:

• decreases
• increases

Because:

5.2 Collapsing Star (Astrophysical Insight)

If stellar radius decreases drastically:

Thus angular velocity increases dramatically.

This explains formation of rapidly spinning neutron stars.

6. Energy Interpretation Under Angular Momentum Conservation

Rotational kinetic energy:

Using :

Substituting into energy expression:

This reveals a very important fact:

If decreases, kinetic energy increases.

Thus angular momentum conservation does not imply energy conservation.

Energy increase comes from internal work done by the system.

This subtle point is frequently tested in JEE Advanced conceptual questions.

7. Coupled Systems and Angular Momentum Transfer

Consider two discs that stick together after collision.

Initial angular momentum:

Final angular momentum:

Using conservation:

Solve for .

This approach is far simpler than torque-based methods.

8. Variable Moment of Inertia Problems

Suppose radius doubles while mass remains constant:

If becomes :

becomes

To conserve angular momentum:

becomes

This type of proportional reasoning is extremely common in objective questions.

9. Angular Momentum and Rotational Equilibrium

If:

Then:

Thus angular velocity remains constant.

But if moment of inertia changes, angular velocity adjusts while preserving .

This distinction is subtle but crucial.

10. Advanced JEE Multi-Step Concept

Consider a rotating platform with a person walking radially inward.

Steps to solve:

1. Identify absence of external torque.
2. Apply conservation of angular momentum.
3. Express new angular velocity using .
4. If required, compute new kinetic energy.

Such problems test both conservation principles and energy interpretation.

11. Important Conceptual Insights

• Angular momentum depends on axis of rotation.
• Only external torque changes total angular momentum.
• Internal forces cancel in pairs.
• Conservation of angular momentum is valid even if energy changes.
• Angular momentum conservation is often more fundamental than energy conservation in rotational problems.

12. Common Errors in Exams

• Applying for non-fixed axis motion.
• Ignoring external torque due to friction or tension.
• Confusing linear and angular momentum conservation.
• Forgetting vector nature in 3D problems.
• Assuming energy conservation when angular momentum is conserved.

Avoiding these mistakes ensures clarity in advanced rotational problems.

FAQs – 6.12 Angular Momentum about a Fixed Axis

Q1. What is angular momentum of a rigid body about a fixed axis?

It is .

Q2. When is angular momentum conserved?

When net external torque acting on the system is zero.

Q3. What is the relation between torque and angular momentum?

.

Q4. Does conservation of angular momentum imply conservation of energy?

No. Energy may change if moment of inertia changes.

Q5. Why does angular velocity increase when radius decreases?

Because must remain constant in absence of external torque.

Conclusion

Angular momentum in case of rotations about a fixed axis is given by:

It changes only when an external torque acts:

When no external torque is present, angular momentum remains conserved:

This conservation principle is one of the most elegant and powerful tools in classical mechanics. A deep understanding of angular momentum, its vector nature, its conservation, and its energy implications completes the conceptual framework of rotational motion in Chapter 6 and prepares students for high-level JEE problem solving.