7.4 Conservation of angular momentum

Conservation of angular momentum is a key principle in Engineering Mechanics – Dynamics. It describes how rotational motion is preserved in systems without external torques, crucial for analyzing spinning objects and orbital motion.

The concept links moment of inertia, angular velocity, and external torques. It's applied in various fields, from ice skating to satellite control, and is fundamental to understanding planetary motion and complex mechanical systems.

Definition of angular momentum

• Angular momentum quantifies rotational motion in physics and engineering
• Plays a crucial role in analyzing rotating systems in Engineering Mechanics – Dynamics
• Provides insights into the behavior of spinning objects and orbital motion

Angular momentum equation

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• Defined mathematically as $L = I\omega$
• $I$ represents the moment of inertia of the rotating object
• $\omega$ denotes the angular velocity of the object
• For a point mass, angular momentum calculated as $L = r \times p$
  • $r$ is the position vector from the axis of rotation
  • $p$ is the linear momentum of the point mass

Units of angular momentum

• Expressed in kg⋅m²/s in SI units
• Derived from the product of moment of inertia (kg⋅m²) and angular velocity (rad/s)
• Imperial units include lb⋅ft²/s
• Dimensionally equivalent to energy multiplied by time

Conservation principle

• Fundamental concept in Engineering Mechanics – Dynamics
• States that angular momentum remains constant in the absence of external torques
• Applies to both rotating and orbiting systems

Isolated systems

• No external torques acting on the system
• Total angular momentum remains constant over time
• Includes closed systems with no interaction with the environment
• Examples include spinning tops in vacuum and planetary systems

External torques

• Change the angular momentum of a system
• Calculated as the time derivative of angular momentum: $\tau = \frac{dL}{dt}$
• Can increase or decrease angular momentum depending on direction
• Examples include friction in mechanical systems and gravitational forces in orbital mechanics

Moment of inertia

• Measures an object's resistance to rotational acceleration
• Analogous to mass in linear motion
• Depends on the mass distribution of the object relative to the axis of rotation

Parallel axis theorem

• Relates moment of inertia about any axis to that about a parallel axis through the center of mass
• Expressed mathematically as $I = I_{cm} + md^2$
  • $I_{cm}$ is the moment of inertia about the center of mass
  • $m$ is the total mass of the object
  • $d$ is the perpendicular distance between the axes
• Useful for calculating moments of inertia for complex shapes

Perpendicular axis theorem

• Applies to planar objects with uniform thickness
• States that the sum of moments of inertia about two perpendicular axes equals the moment of inertia about the axis perpendicular to the plane
• Expressed as $I_z = I_x + I_y$
• Simplifies calculations for symmetric objects (discs, rectangles)

Angular momentum in rotation

• Describes the rotational motion of rigid bodies
• Essential for analyzing spinning objects in engineering applications

Fixed axis rotation

• Rotation occurs around a stationary axis
• Angular momentum remains parallel to the axis of rotation
• Magnitude of angular momentum given by $L = I\omega$
• Examples include wheels, gears, and rotating shafts

General rotation

• Involves rotation about a moving axis
• Described by Euler's equations of motion
• Angular momentum vector can change direction
• Applies to complex motions (gyroscopes, spinning satellites)

Collisions and angular momentum

• Analyzes the transfer of angular momentum during impact events
• Important in studying vehicle crashes, sports mechanics, and particle collisions

Elastic vs inelastic collisions

• Elastic collisions conserve both kinetic energy and angular momentum
• Inelastic collisions conserve angular momentum but not kinetic energy
• Coefficient of restitution determines the degree of elasticity
• Examples include billiard ball collisions (elastic) and car crashes (inelastic)

Impact parameter

• Perpendicular distance between the line of approach and the center of mass
• Determines the amount of angular momentum transferred during a collision
• Large impact parameters result in greater angular momentum transfer
• Critical in analyzing glancing collisions and orbital captures

Applications of conservation

• Demonstrates practical uses of angular momentum conservation in engineering and physics
• Provides insights into the behavior of rotating systems in various fields

Spinning ice skater

• Demonstrates the conservation of angular momentum in human motion
• As the skater pulls their arms in, their moment of inertia decreases
• Angular velocity increases to conserve angular momentum
• Results in faster rotation speed during spins

Satellite attitude control

• Utilizes conservation of angular momentum for spacecraft orientation
• Reaction wheels or control moment gyroscopes change the satellite's angular momentum
• Allows precise pointing of antennas and instruments
• Essential for communication satellites and space telescopes

Gyroscopes

• Exploit conservation of angular momentum to maintain orientation
• Resist changes in their axis of rotation due to high angular momentum
• Used in navigation systems, stabilization platforms, and inertial guidance
• Applications include aircraft instruments and smartphone sensors

Vector nature of angular momentum

• Emphasizes the directional properties of angular momentum
• Critical for understanding three-dimensional rotational motion

Right-hand rule

• Determines the direction of angular momentum vector
• Curl fingers of right hand in direction of rotation
• Thumb points in direction of angular momentum vector
• Applies to both linear and angular velocities

Cross product representation

• Expresses angular momentum as a cross product: $\vec{L} = \vec{r} \times \vec{p}$
• $\vec{r}$ is the position vector from the axis of rotation
• $\vec{p}$ is the linear momentum vector
• Results in a vector perpendicular to both $\vec{r}$ and $\vec{p}$

Angular momentum in orbital motion

• Applies conservation principles to celestial mechanics
• Fundamental in understanding planetary motion and spacecraft trajectories

Kepler's laws

• First law describes elliptical orbits with the sun at one focus
• Second law states that equal areas are swept in equal times
• Third law relates orbital period to semi-major axis
• All laws derived from conservation of angular momentum and energy

Planetary motion

• Planets conserve angular momentum in their orbits around the sun
• Explains why planets move faster when closer to the sun (perihelion)
• Governs the stability of planetary systems
• Applies to both natural and artificial satellites

Relationship to linear momentum

• Connects rotational and translational motion concepts
• Essential for understanding complex mechanical systems

Translation vs rotation

• Linear momentum describes translational motion of center of mass
• Angular momentum characterizes rotational motion about an axis
• Both conserved in isolated systems
• Combined analysis required for general motion of rigid bodies

Center of mass motion

• Angular momentum can be separated into center of mass and relative components
• Total angular momentum includes contribution from center of mass motion
• Useful in analyzing systems of particles or extended bodies
• Applies to rocket propulsion and multi-body dynamics

Conservation in multiple particle systems

• Extends angular momentum concepts to complex systems
• Crucial for analyzing molecular dynamics and astrophysical phenomena

Internal vs external forces

• Internal forces do not change the total angular momentum of the system
• External forces can produce torques that alter total angular momentum
• Conservation applies to internal interactions (collisions, chemical reactions)
• Examples include molecular vibrations and galaxy formation

System boundaries

• Define the region where conservation of angular momentum applies
• Can be physical (container walls) or conceptual (control volume)
• Proper definition crucial for accurate analysis
• Affects the treatment of forces as internal or external

Angular impulse

• Represents the change in angular momentum due to applied torques
• Analogous to linear impulse in translational motion

Impulse-momentum theorem

• States that angular impulse equals change in angular momentum
• Expressed mathematically as $\int \tau dt = \Delta L$
• Applies to both constant and time-varying torques
• Useful for analyzing impacts and sudden changes in rotation

Time-varying torques

• Require integration over time to determine total angular impulse
• Can result in complex changes in angular momentum
• Examples include engine torque curves and wind loading on structures
• Analyzed using numerical methods for non-trivial cases

Numerical methods

• Provide techniques for solving complex angular momentum problems
• Essential for engineering analysis of real-world rotational systems

Finite difference approximations

• Discretize time and space to solve differential equations
• Used to approximate derivatives in equations of motion
• Include forward, backward, and central difference schemes
• Accuracy improves with smaller time steps

Runge-Kutta methods

• Higher-order numerical integration techniques
• Provide better accuracy than simple finite difference methods
• Fourth-order Runge-Kutta (RK4) widely used in engineering simulations
• Balances computational efficiency and accuracy for many applications