KNES 361 L13

Lecture 13: Angular Kinetics and the Law of Conservation of Angular Momentum

Introduction to Angular Momentum

• Angular momentum is defined as the quantity possessed by an object due to its angular motion.

• The key objectives of this lecture:

  • Explore the concept of angular momentum.

  • Introduce the law of conservation of angular momentum.

  • Examine real-world applications of these principles, such as gymnastics, diving, and figure skating.

Comparison with Linear Momentum

• Linear momentum refers to the quantity of motion an object has due to its linear trajectory.

• Formula for linear momentum:
  $p = m imes v$
  where:

  • $p$ = linear momentum

  • $m$ = mass of the object

  • $v$ = linear velocity

• Mass is associated with inertia, which represents resistance to changes in motion.

• It is critical to transition from linear to rotational concepts as we discuss angular momentum.

Angular Momentum

• Angular momentum is the rotational analog of linear momentum.

• Formula for angular momentum: $h = I imes heta$ where:

  • $h$ = angular momentum

  • $I$ = moment of inertia

  • $heta$ = angular velocity

Components of Angular Momentum

• Moment of Inertia ($I$): The rotational equivalent of mass, representing how difficult it is to change an object's rotational motion. Calculation:
  $I = m imes k^2$
  where:

  • $k$ = radius of gyration, which reflects how mass is distributed around the axis of rotation.

• Angular Velocity ($heta$): Indicates how fast an object is spinning.

Angular Momentum Dependence Factors

• Angular momentum depends on:

  • Mass of the object ($m$)

  • Radius of gyration ($k$)

  • Angular velocity ($heta$)

• More specifically:
  $h = m imes k^2 imes heta$

• Key interpretation of the formula:

  • Increased mass results in increased angular momentum.

  • Increased angular velocity also results in increased angular momentum.

  • Larger radius of gyration leads to greater angular momentum, more significantly affecting the angular momentum due to being squared in the equation.

Importance of Radius of Gyration

• Among the three factors affecting angular momentum, the radius of gyration (mass distribution) plays the most significant role because it is squared in the equation.

• If mass is further distributed from the axis, angular momentum increases markedly.

Law of Conservation of Angular Momentum

• The principle states that angular momentum is conserved when no external torque acts on a system.

• When airborne, the only force acting on an object is gravity, which does not apply torque in a way that changes the angular momentum.

• Key takeaway: when in the air, the total spin of an object remains unchanged.

Mathematical Interpretation of Conservation

• Given that angular momentum ($h$) remains constant, the relationship can be deduced:

  • If moment of inertia ($I$) increases, then angular velocity ($heta$) must decrease, and vice versa:
    $I<em>1 imes heta</em>1 = I<em>2 imes heta</em>2$

Application in Sports: Gymnasts and Ice Skaters

• When a gymnast or ice skater tucks their body:

  • Moment of inertia decreases (mass moves closer to the axis).

  • As a result, angular velocity increases, allowing for a faster spin.

Understanding with Examples

• Example: A figure skater begins spinning at 1 revolution per second. If her moment of inertia reduces to 50% of the original value:

  1. Original angular momentum can be represented as:
     $I<em>{ ext{initial}} imes heta</em>{ ext{initial}} = I<em>{ ext{final}} imes heta</em>{ ext{final}}$

  2. The final moment of inertia can be expressed as:
     $I<em>{ ext{final}} = 0.5 imes I</em>{ ext{initial}}$

  3. Therefore:
     $I<em>{ ext{initial}} imes heta</em>{ ext{initial}} = 0.5 imes I<em>{ ext{initial}} imes heta</em>{ ext{final}}$

  4. This simplifies to:
     $heta<em>{ ext{final}} = 2 imes heta</em>{ ext{initial}}$

• Given that the initial angular velocity was 1 revolution per second:

  • The new angular velocity would be:
    $heta_{ ext{final}} = 2 ext{ revolutions per second}$

Conclusion

• The law of conservation of angular momentum explains critical movements in gymnastics, diving, and figure skating.

• Key concepts:

  • Changing body configurations allows a performer to manipulate moment of inertia and subsequently angular velocity while conserving angular momentum.

• Essential to remember that in mid-air, no new angular momentum can be generated; only its configuration can be altered for desired motion.