Lecture Notes: Angular Momentum and Angular Impulse

Introduction

• Discussion on angular momentum, its conservation, and angular impulse.
• XKCD web comic: A clever depiction about spinning counter-clockwise to affect the Earth's rotation slightly.

Angular Momentum

Definition and Equation

• Linear vs Angular Momentum:
  • Linear momentum: ( p = mv )
  • Angular momentum (L): ( L = I \omega )
• Rotational Equivalents:
  • Mass ( \rightarrow ) Moment of inertia ( I )
  • Velocity ( \rightarrow ) Angular velocity ( \omega )

Units

• Angular Velocity: Radians per second
• Moment of Inertia (I): ( \sum mr^2 )
  • Units: kilogram meter squared ( \mathrm{(kg , m^2)} )
• Angular Momentum (L): ( \mathrm{(kg , m^2/s)} )

Conservation of Angular Momentum

• Principle: Total angular momentum is constant if no external net torques act on the system.
• Equation: ( L = I \omega )
• Application:
  • Example of a spinning figure skater reducing radius ( r ) to increase ( \omega ) (spin faster).
  • Demonstration with a video: Figure skater's spinning dynamics.
• Large-Scale Example: Crab Nebula
  • Historical supernova observation by Chinese astronomers in 1054.
  • Formation of the Crab Nebula and presence of a neutron star.
  • Explanation of a pulsar: Neutron star spinning rapidly due to reduced radius.

Angular Impulse

Definition and Equation

• Linear Impulse: ( J = F \Delta T = \Delta p )
• Angular Impulse: ( J = \Delta L )
  • Equivalent to torque ( \times ) change in time ( \Delta T )
  • ( \Delta L = \Delta (I \omega) )

Summary

• Key Concepts:
  • Angular momentum and its conservation.
  • Angular impulse as the rotational equivalent of linear impulse.
• Applications and Implications:
  • From everyday phenomena (e.g., figure skaters) to astronomical events (e.g., supernovae and neutron stars).