Angular Impulse

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 helps indicate the direction that the system will rotate in (clockwise or counterclockwise) since it is associated with change in velocity and acceleration.

The Main Idea

Angular impulse is the torque acting over some time interval, or the change in angular momentum. Angular momentum can be changed by an angular impulse. 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]

The angular impulse is equal to the moment of inertia [math]\displaystyle{ I }[/math] times the change in angular velocity [math]\displaystyle{ \Delta\vec{\omega} }[/math].

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

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

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Connectedness

Angular impulse is present in so many things in daily life, from wheels turning on a bicycle to turning the steering wheel in a car and even a person just spinning around in place. Personally, I'm really interested in computers, desktop computers in particular. This topic relates to the turning of fans in my case, on my graphics card, and on my processor, so angular momentum is critical when it can mean a negative one would result in drastically lower fan speeds that would make a computer overheat or a postive one would result in an increase in fan speed which would likely result in the fans being really noisy and annoying.

As a Computer Science major, angular momentum relates to actual computers in the example I gave previously. Not only that, but in the branch of artificial intelligence, if robots are involved then angular impulse can be critical in their circular motion.

Angular impulse has numerous industrial applications, being critical in any rotating device, like cars (wheels, steering wheels), generators, and even in water/wind mills which can provide hydroelectric/wind power.

History

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See also

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Further reading

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External links

Angular Momentum Impulse Video

References

Chapter 11 of Matter & Interactions 4th Edition

List of Equations in Classical Mechanics

Lesson 13 Angular Impulse