Angular Momentum

Introduction

Key Concepts

Definition of Angular Momentum

Angular momentum, often denoted by the symbol \( \vec{L} \), is a vector quantity that represents the quantity of rotation an object has, taking into account its mass, shape, and speed. It is analogous to linear momentum but applies to rotational motion. Mathematically, for a single particle, angular momentum is defined as:

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

where:

• \( \vec{r} \) is the position vector of the particle relative to a chosen origin.
• \( \vec{p} = m\vec{v} \) is the linear momentum of the particle.

For a rigid body rotating about a fixed axis, the angular momentum can be expressed as:

$$ \vec{L} = I\vec{\omega} $$

where:

• \( I \) is the moment of inertia of the object.
• \( \vec{\omega} \) is the angular velocity vector.

Moment of Inertia

The moment of inertia \( (I) \) is a scalar measure of an object's resistance to changes in its rotation. It depends on the mass distribution relative to the axis of rotation. For different shapes and mass distributions, the moment of inertia varies. Some common moments of inertia include:

• Solid Cylinder or Disk about its central axis: \( I = \frac{1}{2}MR^2 \)
• Hollow Cylinder or Thin Hoop about its central axis: \( I = MR^2 \)
• Solid Sphere about its diameter: \( I = \frac{2}{5}MR^2 \)
• Thin Spherical Shell about its diameter: \( I = \frac{2}{3}MR^2 \)

The general formula for the moment of inertia for a system of particles is:

$$ I = \sum_{i} m_i r_i^2 $$

where \( m_i \) is the mass of the \( i^{th} \) particle and \( r_i \) is its distance from the axis of rotation.

Angular Velocity and Angular Acceleration

Angular velocity \( (\omega) \) is the rate of change of angular displacement and is measured in radians per second (rad/s). It describes how quickly an object rotates or revolves relative to another point. Angular acceleration \( (\alpha) \) is the rate of change of angular velocity and measures how quickly an object is speeding up or slowing down its rotation.

The relationship between angular displacement \( (\theta) \), angular velocity \( (\omega) \), and angular acceleration \( (\alpha) \) can be summarized as:

• \( \omega = \frac{d\theta}{dt} \)
• \( \alpha = \frac{d\omega}{dt} \)

Conservation of Angular Momentum

In the absence of external torques, the total angular momentum of a system remains constant. This principle is known as the conservation of angular momentum and is pivotal in various physical phenomena such as the spinning of ice skaters and the behavior of celestial bodies.

Mathematically, if no external torque (\( \tau_{\text{ext}} \)) acts on a system, then:

$$ \frac{d\vec{L}}{dt} = \vec{\tau}_{\text{ext}} = 0 \Rightarrow \vec{L} = \text{constant} $$

This implies that any change in the distribution of mass or the rotation rate must compensate to keep angular momentum unchanged.

Torque

Torque \( (\tau) \) is the rotational equivalent of force and measures the tendency of a force to rotate an object about an axis. It is a vector quantity defined as the cross product of the position vector and the force vector:

$$ \vec{\tau} = \vec{r} \times \vec{F} $$

Where:

• \( \vec{r} \) is the position vector from the axis of rotation to the point where the force is applied.
• \( \vec{F} \) is the force vector.

The magnitude of torque is given by:

$$ \tau = rF\sin(\theta) $$

where \( \theta \) is the angle between \( \vec{r} \) and \( \vec{F} \).

Angular Impulse

Angular impulse is the product of torque and the time over which it acts. It results in a change in angular momentum:

$$ \text{Angular Impulse} = \vec{\tau} \Delta t = \Delta \vec{L} $$

This equation is analogous to the linear impulse-momentum theorem and highlights how torque affects the rotational motion of an object over time.

Applications of Angular Momentum

Angular momentum has a wide array of applications in both natural and engineered systems. Some notable applications include:

• Astronomy: The conservation of angular momentum explains the formation of planets and stars from collapsing gas clouds.
• Sports: Athletes use angular momentum to control their spins, such as figure skaters pulling in their arms to spin faster.
• Engineering: Gyroscopes utilize angular momentum to maintain orientation in navigation systems.
• Everyday Objects: Devices like electric drills and washing machines operate based on principles of angular momentum.

Mathematical Derivations and Examples

To better understand angular momentum, let's explore a couple of examples demonstrating its calculation and conservation.

Example 1: Calculating Angular Momentum of a Rotating Disk

Consider a solid disk with mass \( M = 2.0 \, \text{kg} \) and radius \( R = 0.5 \, \text{m} \) rotating with an angular velocity \( \omega = 10 \, \text{rad/s} \). Calculate its angular momentum.

First, determine the moment of inertia \( I \) for a solid disk:

$$ I = \frac{1}{2}MR^2 = \frac{1}{2}(2.0 \, \text{kg})(0.5 \, \text{m})^2 = 0.25 \, \text{kg} \cdot \text{m}^2 $$

Now, calculate the angular momentum \( L \):

$$ L = I\omega = 0.25 \, \text{kg} \cdot \text{m}^2 \times 10 \, \text{rad/s} = 2.5 \, \text{kg} \cdot \text{m}^2/\text{s} $$

Example 2: Conservation of Angular Momentum in a Figure Skater's Spin

A figure skater spinning with arms extended has an angular momentum of \( L = 30 \, \text{kg} \cdot \text{m}^2/\text{s} \). If she pulls her arms in, reducing her moment of inertia by a factor of 2, what is her new angular velocity?

Using the conservation of angular momentum \( (L_{\text{initial}} = L_{\text{final}}) \):

$$ L_{\text{initial}} = I_{\text{initial}} \omega_{\text{initial}} = I_{\text{final}} \omega_{\text{final}} $$

Given \( I_{\text{final}} = \frac{1}{2}I_{\text{initial}} \), we have:

$$ I_{\text{initial}} \omega_{\text{initial}} = \frac{1}{2}I_{\text{initial}} \omega_{\text{final}} \Rightarrow \omega_{\text{final}} = 2\omega_{\text{initial}} $$

Thus, her angular velocity doubles when she pulls her arms in.

Derivation of Angular Momentum for a Rigid Body

For a rigid body rotating about a fixed axis, the total angular momentum is the sum of the angular momenta of all its particles. Starting with:

$$ \vec{L} = \sum_{i} \vec{r}_i \times \vec{p}_i = \sum_{i} \vec{r}_i \times m_i\vec{v}_i $$

Since \( \vec{v}_i = \vec{\omega} \times \vec{r}_i \), substitute into the equation:

$$ \vec{L} = \sum_{i} \vec{r}_i \times m_i (\vec{\omega} \times \vec{r}_i) $$

Using the vector triple product identity \( \vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}(\vec{a} \cdot \vec{c}) - \vec{c}(\vec{a} \cdot \vec{b}) \), we simplify:

$$ \vec{L} = \vec{\omega} \sum_{i} m_i r_i^2 = I\vec{\omega} $$

Thus, the angular momentum of a rigid body is directly proportional to its angular velocity, with the moment of inertia as the proportionality constant.

Applications in Real-World Scenarios

Understanding angular momentum is essential in analyzing and designing systems where rotational motion is involved. Here are some real-world applications:

• Spacecraft Attitude Control: Angular momentum is used to control the orientation of spacecraft without expending fuel, utilizing devices like reaction wheels and control moment gyroscopes.
• Bicycle Stability: The spinning wheels of a bicycle generate angular momentum, contributing to the bicycle's stability during motion.
• Rotational Machines: Engines, turbines, and other machinery rely on angular momentum principles to function efficiently.

Comparison Table

Aspect	Angular Momentum	Linear Momentum
Definition	Quantity of rotation of a body, dependent on mass, shape, and speed.	Product of mass and velocity of an object.
Formula	\( \vec{L} = \vec{r} \times \vec{p} \)	\( \vec{p} = m\vec{v} \)
Conservation Principle	Conserved in absence of external torque.	Conserved in absence of external force.
Units	kg.m²/s	kg.m/s
Application	Rotational dynamics of planets, spinning tops, and machinery.	Motion of vehicles, projectiles, and fluid dynamics.
Relation to Torque	Change in angular momentum is caused by torque.	Change in linear momentum is caused by force.

Summary and Key Takeaways