Angular Momentum in Rotational Motion - Notes for Quick Revision 📚

Angular Momentum in Rotational Motion - Rotational motion is a key concept in physics that deals with objects spinning around an axis. One of the most important quantities in rotational dynamics is angular momentum, which is the rotational analog of linear momentum. Angular momentum describes the quantity of rotation an object possesses and how it interacts with external torques.

This article provides comprehensive notes on angular momentum, covering its definition, formula, principles, theorems, applications, and examples.

What is Angular Momentum?

Definition: Angular momentum is a measure of the rotational motion of an object around a fixed axis.

Formula:

$$L=I\omega$$ Where:

• $L$: Angular momentum (measured in ${\text{kg}\text{cdotp}\text{m}}^2\mathrm{/}\text{s}$)
• $I$: Moment of inertia of the object about the axis of rotation (${\text{kg}\text{cdotp}\text{m}}^2$).
• $\omega$: Angular velocity ($\text{rad/s}$).

Angular Momentum for a Particle

For a single particle moving in a plane:

$$L = r \times p$$

Where:

• $r$: Position vector of the particle relative to the axis of rotation.
• $p$: Linear momentum ($p = mv$).
• $\times$: Represents the cross product.
• Magnitude: $$L = r m v \sin\theta$$
  • $\theta$: Angle between $r$ and $v$.

Key Characteristics of Angular Momentum

Vector Quantity:

• Angular momentum has both magnitude and direction.
• Its direction is determined using the right-hand rule.

SI Unit:

• ${\text{kg}\text{cdotp}\text{m}}^2\mathrm{/}\text{s}$

Dependence:

• Directly proportional to the mass, distance from the axis of rotation, and angular velocity.

Angular Momentum of a System of Particles

For a system of particles, the total angular momentum is the vector sum of the angular momentum of all individual particles:

$$L_{\text{total}} = \sum L_i$$

Where $L_i = r_i \times p_i$ for each particle $i$.

Relation Between Torque and Angular Momentum

Torque is the rate of change of angular momentum:

$$\tau = \frac{dL}{dt}$$

Where:

• $\tau$: Net external torque.
• $dL/dt$: Time rate of change of angular momentum.

If the net external torque is zero, angular momentum remains constant, which is the basis of the law of conservation of angular momentum.

Law of Conservation of Angular Momentum

Statement:

The angular momentum of a system remains constant if no external torque acts on the system.

Mathematical Form:

$$I_1\omega_1 = I_2\omega_2$$

Where:

• $I_1, I_2$: Moments of inertia in two states.
• $\omega_1, \omega_2$: Angular velocities in two states.

Applications of Conservation of Angular Momentum

Astrophysics:

• A collapsing star spins faster as it contracts, conserving angular momentum (e.g., formation of neutron stars).

Figure Skating:

• A skater spins faster by pulling their arms close to their body, reducing $I$ and increasing $\omega$.

Rotating Machines:

• Flywheels in engines rely on angular momentum conservation for stability.

Earth’s Rotation:

• The conservation of angular momentum explains phenomena like the precession of Earth's axis.

Angular Momentum in Terms of Rotational Kinetic Energy

Rotational kinetic energy ($KE_{\text{rot}}$) is related to angular momentum:

$$KE_{\text{rot}} = \frac{L^2}{2I}$$

Where:

• $L$: Angular momentum.
• $I$: Moment of inertia.

Difference Between Angular Momentum and Linear Momentum

Angular Momentum	Linear Momentum
Measures rotational motion.	Measures linear motion.
Depends on $I$ and $\omega$.	Depends on $m$ and $v$.
SI Unit: $\text{kg·m}^2/\text{s}$.	SI Unit: $\text{kg·m/s}$.

Examples of Angular Momentum

Rotating Wheel:

• A spinning bicycle wheel has angular momentum due to its rotation.

Planetary Motion:

• Planets conserve angular momentum as they orbit the Sun, moving faster when closer and slower when farther.

Pirouette in Ballet:

• A dancer spinning faster by tucking their arms demonstrates conservation of angular momentum.

Angular Momentum Theorems

Parallel Axis Theorem:

• The moment of inertia about any axis parallel to an axis through the center of mass is given by: $$I=I_{\text{cm}}+M{d}^2$$ Where:
  • $I_{\text{cm}}$: Moment of inertia about the center of mass.
  • $M$: Mass of the object.
  • $d$: Distance between the axes.

Perpendicular Axis Theorem (for planar bodies):

• The moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia about two perpendicular axes in the plane: $$I_z=I_x+I_y$$

FAQs About Angular Momentum

What does angular momentum represent?

Angular momentum represents the rotational equivalent of linear momentum, describing the quantity of rotation an object possesses.

How is angular momentum conserved?

Angular momentum is conserved when no external torque acts on a system.

Why does a skater spin faster when they pull their arms in?

Pulling arms in decreases the moment of inertia ($I$), causing angular velocity ($\omega$) to increase to conserve angular momentum.

How does angular momentum relate to torque?

Torque is the time rate of change of angular momentum ($\tau = dL/dt$).

Can angular momentum be negative?

Yes, angular momentum can be negative depending on the direction of rotation.

Angular momentum is a cornerstone of rotational dynamics, playing a pivotal role in physics, engineering, and natural phenomena. From the spinning Earth to high-speed turbines, understanding angular momentum and its conservation helps us analyze and predict rotational behaviors. These notes provide a complete understanding of angular momentum, its principles, and applications, making them an invaluable resource for academic and competitive exams.

Explore related topics:

• Rotational Motion and Moment of Inertia
• Torque in Rotational Motion
• Kinematics