I am doing some physics for a game.
I have a rigid body defined as multiple balls of the same mass distributed to make some object.
Let's say I put to each ball a different velocity but because it is a rigid body the only thing that matters in the end is its angular velocity and the velocity at the center of mass, I know how to get the velocity at the center of mass which is summing all the balls velocities and divide it by the number of balls, but I cannot figure out how to mix all the velocities to get the angular velocity of the whole body.
[Physics] angular velocity of rigid body
angular velocity
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Does this mean we can somehow add the 3 rotations (which are about different axes) and get an equivalent rotation about some other (single) axis?
Yes. It's one of many consequences of Euler's rotation theorem. This neat little trick of finding the Euler rotation axis works in three dimensional space, and in three dimensional space only. The Lie group SO(3) is a very special space.
A more generic way of looking at rotation is that rotation in any Euclidean space comprises a combinations of rotations parallel to a two dimensional plane rather than about an axis. There's only one plane in two dimensional space, so rotation in 2D space requires but one parameter. There are three pairs of planes of rotation in three dimensional space (the YZ, ZX, and XY planes), so rotation in 3D space requires three parameters. You can think of those planar rotations as being about an axis (with an arbitrary decision as to what qualifies as positive or negative rotation). There are six planes of rotation in four dimensional space, so rotations in 4D space requires six parameters. This can result in something very weird, something you don't see in 2D or 3D space, a Clifford rotation.
It has already been somewhat answered in this other question of yours: Parallel axis theorem and Koenig theorem for angular momentum (see also my comment to the answer).
Angular velocity, like translational velocity, has to be defined by reference to a frame. In your question, you state the angular velocity about the principal axis is $ω$. So its angular velocity is... $ω$. If you study the motion in the (non galilean) frame centered on the CM and having an axis pointing inwards to $z$, the angular velocity will be $ω-Ω$. This is reminiscent of the difference between stellar day and solar day.
Moment of inertia is used for rotation around a point or axis. $I_z$ is as you say; but this would be of use only if the body were rotating around $z$. Which is not the case. The movement here is a combination of:
- rotation about the principal axis,
- circular translation (called revolution in astronomy) about $z$.
To compute angular momentum or kinetic energy in combined motions like this one, use König's theorems.
Best Answer
You are on the right track with your line of thinking, but as per our comments above, it may be better to think about applying forces to each ball as opposed to velocities. If you apply a force to each ball you can find the center of mass motion by summing all of the forces on all of the balls and using $\vec{F}_{net}=m_{tot}\vec{a}_{CM}$, where $F_{net}$ is the vector sum of all of the forces, $m_{tot}$ is the total mass and $a_{CM}$ is the acceleration of the center of mass.
To find the rotation of the body, you need to find the net torque about your axis of rotation. If there is no fixed axis of rotation that you set, the axis of rotation will be about the center of mass. For each ball, you can calculate the torque from the force applied from the equation $\vec{\tau} = \vec{r}\times\vec{F}$, where $\vec{r}$ is the vector from the center of mass to the ball in question (or from a different fixed axis of rotation), and $\vec{F}$ is the force applied to that ball. Note that the torque is also a vector, and points along the axis of rotation. To find the net torque, you can do a vector sum of all of the individual torques.
Once you find the net torque, you can then find the net angular acceleration $\alpha$ of the rigid body, by using the formula $\tau = I\alpha$, where $I$ is the momentum of inertia of your rigid body, which can be calculated using $I = \Sigma m_i r_i^2$, where this is a sum of the mass of each ball times the square of the distance from the axis of rotation (center of mass most likely).
The angular velocity of each ball is then given by the angular acceleration times the amount of time that the forces are applied to your object, $\omega = \alpha t$