1. Energy and Momentum of Rotating Systems

1.1 Conservation of Angular Momentum

1.1.1 Conservation of Angular Momentum

1.1.2 Angular Momentum of a System

1.2 Rotational Kinetic Energy, Torque & Work

1.2.1 Rotational Kinetic Energy

1.2.2 Torque & Work

1.3 Examples of Rotational Systems

1.3.1 Rolling

1.3.2 Motion of Orbiting Satellites

1.3.3 Escape Velocity

1.4 Angular Momentum & Angular Impulse

1.4.1 Angular Momentum

1.4.2 Angular Impulse

1.4.3 Impulse–Momentum Theorem in Rotational Form

1.4.4 Angular Impulse Graphs

2. Force and Translational Dynamics

2.1 Gravitational Force

2.1.1 Gravitational Field Strength Equation

2.1.2 Weight

2.1.3 Apparent Weight

2.1.4 Newton's Law of Gravitation

2.1.5 Inertial vs Gravitational Mass

2.1.6 Gravitational Force

2.1.7 Gravitational Field

2.1.8 Acceleration Due to Gravity

2.2 Newton's Laws

2.2.1 Newton's Second Law

2.2.2 Newton's Third Law

2.2.3 Newton's First Law

2.3 Systems & Center of Mass

2.3.1 Describing Systems

2.3.2 Center of Mass

2.4 Kinetic & Static Friction

2.4.1 Static Friction Force Formula

2.4.2 Slipping & Sliding

2.4.3 Kinetic Friction

2.4.4 Kinetic Friction Force Equation

2.4.5 Static Friction

2.5 Circular Motion

2.5.1 Kepler's Third Law

2.5.2 Uniform Circular Motion

2.5.3 Acceleration in Circular Motion

2.5.4 Circular Motion in a Vertical Loop

2.5.5 Circular Motion Examples

2.6 Tension

2.6.1 Tension in Ideal & Non-Ideal Strings

2.7 Spring Forces

2.7.1 Hooke's Law

2.8 Forces & Free-Body Diagrams

2.8.1 Free Body Diagrams

2.8.2 Forces as Interactions

2.8.3 Net Force

2.8.4 Translational Equilibrium

3. Fluids

3.1 Fluid Dynamics

3.1.1 Examples of Fluid Dynamics in Real Life

3.1.2 Analyzing Energy Conservation in Fluids

3.1.3 Bernoulli’s Principle and Its Applications

3.2 Fluid Mechanics

3.2.1 Fluid Velocity

3.2.2 Buoyant Force

3.2.3 Fluid Flow Rates

3.2.4 Continuity Equation

3.2.5 Bernoulli’s Equation

3.2.6 Torricelli’s Theorem

3.3 Fluids and Forces

3.3.1 Fluids in Motion and Newton’s Laws

3.3.2 Interactions Between Fluids and Solids

3.3.3 Analyzing Forces in Fluid Systems

3.3.4 Real-World Examples of Fluid Forces

3.4 Density & Pressure

3.4.1 Fluid Properties

3.4.2 Definition of Pressure

3.4.3 Fluid Pressure Equations

4. Work, Energy and Power

4.1 Translational Kinetic Energy & Potential Energy

4.1.1 Gravitational Potential Energy Between Objects

4.1.2 Translational Kinetic Energy

4.1.3 Potential Energy

4.1.4 Elastic Potential Energy

4.1.5 Gravitational Potential Energy Near a Surface

4.2 Work-Energy Theorem

4.2.1 Work-Energy Theorem

4.3 Work

4.3.1 Defining Work & Work Done

4.3.2 Calculating Work Done

4.4 Conservation of Energy

4.4.1 Conservation of Energy

4.5 Power

4.5.1 Calculating Power

4.5.2 Instantaneous Power

5. Torque and Rotational Dynamics

5.1 Newton’s First & Second Law in Rotational Form

5.1.1 Newton’s Second Law in Rotational Form

5.1.2 Rotational Equilibrium

5.1.3 Newton’s First Law in Rotational Form

5.2 Rotational Kinematics

5.2.1 Rotational Kinematics Graphs

5.2.2 Rotating Systems

5.2.3 Angular Displacement

5.2.4 Angular Velocity

5.2.5 Angular Acceleration

5.2.6 Connecting Linear & Rotational Motion

5.2.7 Rotational Kinematics Equations

5.3 Torque

5.3.1 Defining Torque

5.3.2 Force Diagrams & Torque

5.4 Rotational Inertia

5.4.1 Rotational Inertia

5.4.2 Parallel Axis Theorem

6. Kinematics

6.1 Displacement, Velocity & Acceleration

6.1.1 Velocity

6.1.2 Acceleration

6.1.3 Displacement

6.2 Representing Motion

6.2.1 Kinematic Equations

6.2.2 Motion Graphs

6.2.3 Reference Frames & Relative Motion

6.3 Scalars & Vectors

6.3.1 Scalar & Vector Quantities

6.3.2 Combining Vectors

6.4 Vectors & Motion

6.4.1 Vectors & Motion in Two Dimensions

6.4.2 Projectile Motion

7. Oscillations

7.1 Representing and Analyzing SHM

7.1.1 Features of SHM

7.1.2 Graphical Representation of SHM

7.2 Energy of Simple Harmonic Oscillators

7.2.1 Total Energy of SHM

7.2.2 Kinetic Energy & Potential Energy of SHM

7.3 Simple Harmonic Motion

7.3.1 Defining Simple Harmonic Motion

7.3.2 Frequency & Period of SHM

8. Linear Momentum

8.1 Conservation of Linear Momentum

8.1.1 Elastic & Inelastic Collisions

8.1.2 The Principle of Conservation of Momentum

8.1.3 Momentum of a System

8.1.4 Momentum in Collisions

8.1.5 Momentum in Explosions

8.2 Linear Momentum & Impulse

8.2.1 Impulse Equation

8.2.2 Impulse–Momentum Theorem

8.2.3 Impulse Graphs

8.2.4 Linear Momentum

8.2.5 Rate of Change of Momentum

Angular Momentum

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Angular Momentum

Introduction

Angular momentum is a fundamental concept in physics that describes the rotational motion of objects. In the context of Collegeboard AP Physics 1: Algebra-Based, understanding angular momentum and angular impulse is crucial for analyzing the dynamics of rotating systems. This topic not only builds the foundation for more advanced studies in mechanics but also has practical applications in various scientific and engineering fields.

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

Angular momentum $ (\vec{L}) $ quantifies rotational motion, pivotal for analyzing rotating systems.
The moment of inertia $ (I) $ plays a crucial role in determining an object's resistance to changes in its rotation.
Conservation of angular momentum governs phenomena from spinning skaters to celestial mechanics.
Torque $ (\vec{\tau}) $ is essential for understanding how forces affect rotational motion.
Angular impulse links torque and the change in angular momentum over time.

Examiner Tip

Tips

To master angular momentum for the AP exam, remember the acronym LIM: L is for Angular Inertia, and M stands for Angular Momentum. Use the right-hand rule to determine the direction of vectors. Practice solving problems involving both single objects and systems of particles to strengthen your understanding. Additionally, always check which moment of inertia formula applies to the given shape to avoid calculation errors.

Did You Know

Angular momentum isn't just a physics concept—it's what keeps spinning objects like ice skaters balanced and stable. When a skater pulls in their arms, they spin faster due to the conservation of angular momentum. Additionally, planets maintain their orbits thanks to the angular momentum from their formation billions of years ago. Even satellites rely on angular momentum to stay oriented in space, using devices like gyroscopes to manage their rotation.

Common Mistakes

One common mistake is confusing angular momentum with linear momentum, leading to incorrect applications of formulas. For example, students might use $ p = mv $ instead of $ L = I\omega $ for rotating objects. Another frequent error is miscalculating the moment of inertia by using the wrong formula for the object's shape, such as applying the solid sphere formula to a hollow cylinder. Lastly, overlooking the direction of torque vectors can result in sign errors when using the right-hand rule.

FAQ

What is angular momentum?

Angular momentum is a measure of the amount of rotation an object has, taking into account its mass, shape, and speed. It is given by $ \vec{L} = I\vec{\omega} $.

How is angular momentum conserved?

Angular momentum is conserved in a system with no external torques. This means the total angular momentum before any event is equal to the total angular momentum after.

What is the moment of inertia?

The moment of inertia $ (I) $ quantifies an object's resistance to changes in its rotation and depends on the mass distribution relative to the axis of rotation.

How do you calculate torque?

Torque $ (\tau) $ is calculated using $ \vec{\tau} = \vec{r} \times \vec{F} $, where $ \vec{r} $ is the position vector and $ \vec{F} $ is the force applied.

Can angular momentum be zero?

Yes, angular momentum can be zero if an object is not rotating or if the rotational motions cancel each other out in a system.

What are some real-world applications of angular momentum?

Real-world applications include the stability of bicycles, the operation of gyroscopes in navigation systems, the spinning of celestial bodies, and the control of spacecraft orientation.