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7.8: Angular Momentum Conservation Model Summary

Just as we did for linear momentum conservation, we will summarize the main ideas of the angular momentum conservation model/approach by listing the

  1. constructs, i.e., the “things” or ideas that are get “used” in the model,
  2. the relationships–in mathematical or sentence form– that connect the constructs in meaningful ways, and
  3. the ways of representing the relationships.

Developing a deep and rich understanding of the relationships in a model/approach comes slowly. It is absolutely not something you can memorize. This understanding comes only with repeated hard mental effort over a period of time. A good test you can use to see if you are “getting it” is whether you can tell a full story about each of the relationships. It is the meaning behind the equations, behind the simple sentence relationships, that is important for you to acquire. With this kind of understanding, you can apply a model/approach to the analysis of phenomena you have not thought about before. You can reason with the model

Listed here are the major, most important constructs, relationships, and representations of the angular momentum conservation model.

Constructs

  • Angular Velocity, \(\omega\)
  • Rotational Inertia, \(I\)
  • I Angular Momentum, \(L\) 
  • Net Torque, \(\Sigma\tau\)
  • Angular Impulse, \(angJ\)
  • Newton’s 3rd law
  • Conservation of angular momentum

Relationships

The angular velocity is the time derivative of the displacement:

\[ \omega = \frac{d\theta}{dt} \: or \: \omega_{average} = \frac{\Delta\theta}{\Delta t} \]

The angular momentum of an object measured about some fixed axes is simply the product of the object’s rotational inertia and angular velocity:

\[ L = I\omega \]

The angular impulse of the total (or net) external torque acting on an object equals the product of the average torque and the time interval during which the torque acted.

\[ Net \: Angular \: Impulse_{ext} = angJ = \Sigma\tau_{avge \: ext}\Delta t = \int \Sigma\tau_{ext}(t)dt \]

The directions of torque, impulse, angular velocity, and angular momentum as determined by the right-hand rule.

The torque (angular impulse) exerted by object A on object B is equal and opposite to the torque (angular impulse) exerted by object B on object A.

\[\tau_{A\:on\:B}=-\tau_{B\:on\:A} \: and \ angJ_{A\:on\:B} = -angJ_{B\:on\:A} \]

Conservation of Angular Momentum

If the net external angular impulse acting on a system is zero, then there is no change in the total angular momentum of that system; otherwise, the change in angular momentum is equal to the net external angular impulse.

\[ Net \: Angular \: Impulse_{ext} = angJ= \int \Sigma\tau_{ext}(t)dt = L_{f} - L{i} = \Delta L_{system} \]

Representations

Graphical representation of all vector quantities and (vector relationships) as arrows whose length is proportional to the magnitude of the vector and whose direction is in the direction of the vector quantity.

Algebraic vector equations. Vectors denoted as bold symbols or with small arrows over the symbol.

Component algebraic equations, one equation for each of the three independent directions. A useful way to organize and use the representations of the various quantities that occur in phenomena involving angular momentum, change in angular momentum, and angular impulse and torques is an angular momentum chart, which is totally analogous to the linear momentum chart. The angular momentum chart helps us keep track of what we know about the interaction, as well as helping us see what we don’t know.

The boxes are to be filled in with scaled arrows representing the various angular momenta and changes in angular momenta.

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