Rigid Body Collision: Angular Velocity Question

[Jaevko]

So I'm upgrading my Physics Engine's Collision response system (don't worry, this is a pure physics question, no programming involved! Also don't worry about the time of collision, my collision detection system is separate and returns a time of collision for the response system to work with) and I found this really great guide: http://www.myphysicslab.com/collision.html

The equation I'm struggling with is Vap1 = Va1 + (ωa1 x rap)

Two questions:
1) The angular velocity is and has to be in radians/time_unit, correct? (Probably a stupid question, but I can't figure it out for sure and wanted to be certain). Or does it even matter as long as I'm consistent? Why?
2) How can you cross the angular velocity with the rap vector? Isn't the angular velocity a scalar? An example of how to accomplish the math would be greatly appreciated.

Thank you in advance for your time and help!

 

[Simon Bridge]

1. yes (radiens per sec). yes (it matters). (why?) because if you mess with the units you have to add in a scale factor to convert. (The equation uses the definition of "radians".)

2. Angular velocity is a pseudoector like torque. eg. picture a small mass moving in a circle anticlockwise with radius r and speed v.
At 12O'clock, the instantaneous velocity is to the left, the position vector points straight up, and the angular velocity (pseudo)vector points towards you. Notice how the relationship follows?

Also, in general, the 3D moment of inertia is a tensor.

 

[Jaevko]

Thank you for your reply; that was exactly what I was looking for!

 

[Simon Bridge]

No worries... happy hacking.

 

[PJC01]

1) Yes, the angular velocity is typically measured in radians per unit of time (usually seconds). This is because angular velocity is a measure of how much an object is rotating, and radians are a unit of measurement for angles. It is important to be consistent in your units when working with equations to ensure accurate results.

2) The angular velocity is actually a vector, not a scalar. It has both magnitude (speed of rotation) and direction (axis of rotation). The cross product between the angular velocity vector and the rap vector results in a new vector that is perpendicular to both vectors. This vector represents the linear velocity at the point of contact due to the rotation of the rigid body. Here is an example of how to calculate this mathematically:

Let's say the angular velocity vector is ωa1 = [0, 0, 1] (indicating rotation around the z-axis) and the rap vector is rap = [1, 0, 0] (indicating a point on the x-axis).

To calculate the cross product, we use the following formula: Vap1 = ωa1 x rap = [ωa1y * rapz - ωa1z * rapy, ωa1z * rapx - ωa1x * rapz, ωa1x * rapy - ωa1y * rapx]

Plugging in our values, we get: Vap1 = [0 * 0 - 1 * 0, 1 * 0 - 0 * 0, 0 * 0 - 0 * 1] = [0, 0, 0]

This means that the linear velocity at the point of contact is zero, as expected since the point is not moving due to the rotation.

I hope this helps and good luck with your physics engine!