Angular Impulse

Claimed by Katherine Delgado.

Angular impulse represents the effect of a moment of force, or torque ([math]\displaystyle{ \tau }[/math]), acting on a system over a certain period of time ([math]\displaystyle{ \Delta t }[/math]). Angular impulse indicates the direction that the system will rotate in (clockwise or counterclockwise).

The Main Idea

Angular impulse is the torque acting over some time interval, or the change in angular momentum. If it is positive, it results in the system rotating in a counterclockwise direction. If it is negative, the system will rotate in a clockwise direction. There is no common symbol for angular momentum like how [math]\displaystyle{ \vec{F} }[/math] is for force and [math]\displaystyle{ \vec{p} }[/math] is for momentum, and as a result it is almost always referred to as [math]\displaystyle{ \Delta\vec{L} }[/math], since it is equal to the change in angular momentum [math]\displaystyle{ \vec{L} }[/math], just like how linear impulse ([math]\displaystyle{ J }[/math]) is equal to the change in linear momentum, [math]\displaystyle{ \Delta\vec{p} }[/math].

A Mathematical Model

The angular impulse is equal to the net cross product of a force vector, [math]\displaystyle{ \vec{F} }[/math], applied at a particular location a vector distance [math]\displaystyle{ \vec{d} }[/math] from a pivot point times a specified time interval [math]\displaystyle{ \Delta t }[/math]. This is also equal to the net torque [math]\displaystyle{ \sum{\vec{\tau}} }[/math] times a specified time interval [math]\displaystyle{ \Delta t }[/math].

[math]\displaystyle{ \Delta \vec{L} = \sum{(\vec{F}\times\vec{d})}⋅\Delta t = \sum{\vec{\tau}}⋅\Delta t }[/math]

[math]\displaystyle{ \Delta L = I\Delta\omega = I\omega_f - I\omega_i }[/math]

Angular Momentum Principle

The angular momentum principle directly involves angular impulse as shown in the image below:

Both sides are equal to the net angular impulse for a system.

Units

The units for angular impulse are the same as those for angular momentum: [math]\displaystyle{ kg⋅m^2/s }[/math].

A Computational Model

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External links

[1]

References

Chapter 11 of Matter & Interactions 4th Edition

List of Equations in Classical Mechanics

Lesson 13 Angular Impulse