# Finding Direction of Angular Velocity

Suppose I have a 3D rigid object on which some external forces act at various points located on it. The resulting motion would, in general, be the translation of center of mass plus rotation about the axis passing through Center of Mass. Velocity of center of mass can be easily found from summing external forces, so I have information about 'translational part'. I need to find the direction of Angular velocity or the direction of axis of rotation, assuming I know only about the location on the body where external forces act.

Suppose I find the torque about the center of mass (I chose COM because it simplifies the equations, no need to add pseudo torques).

But the direction of torque doesn't always point towards the axis of rotation (if it does then a nice proof using vectors would be a great help because I haven't been able to prove it), only the component of torque along the axis is proportional to magnitude of Angular acceleration, but that doesn't help because I want to know the direction of axis itself.

Finding angular momentum won't help either because angular momentum vector doesn't always point in the direction of angular velocity.

So if I know about the forces acting on the rigid body then how to predict the direction of rotational axis?

Basically knowing the direction of rotational axis passing through COM is important as once known I can simply take the component of net torque (about COM) along the axis to find angular acceleration and to fully analyze the motion.

The sum of torques on the center of mass $$\vec{\tau}_C$$ will let you know the direction of the angular acceleration $$\vec{\alpha}$$, by solving the Euler equations of motion $$\vec{\tau}_C = \mathbf{I}_C \vec{\alpha} + \vec{\omega} \times \mathbf{I}_C \vec{\omega}$$

The result depends on the current angular momentum vector $$\vec{L}_C = \mathbf{I}_C \vec{\omega}$$

Note that the 3×3 mass moment of inertia tensor is calculated from the body $$\mathbf{I}_{\rm body}$$ using the 3×3 rotation matrix $$\mathbf{R}$$ that transforms from local directions to world directions, as follows

$$\mathbf{I}_C = \mathbf{R}\, \mathbf{I}_{\rm body} \mathbf{R}^\top$$

As far as the future angular velocity $$\vec{\omega}$$, you have to integrate the rotational acceleration over time to find the result. Remember a force does to result with velocity, but with acceleration. And acceleration over time results in velocity. Similarly for rotational dynamics.

Have you come across Inertia Tensors? If I've understood your question, then maybe take a look at how these work: 3D Rigid Body Dynamics: The Inertia Tensor.

These govern the mathematics of the direction of angular momentum and angular velocity.

• Unfortunately, no I don't know about Inertia Tensors. But I will take a look. – Shivansh J Aug 23 at 10:26
• Is my question difficult to understand? Basically what I'm asking is a way to find the direction of axis of rotation. Sometimes it is intuitively clear example you have a disc rolling on a flat surface, axis of rotation is perpendicular to the surface of the disc. But what is the most general method to find it? Because in 3D intuitive ability doesn't help anymore. – Shivansh J Aug 23 at 10:30
• What do you know about angular velocity, $\omega$, and its unit vector? Look at how $\omega$ is derived, that may give you some clues about finding the direction of the axis of rotation. – Pox 219 Aug 23 at 10:53

In general, the vector sum of the external torques equals dL/dt, the rate of change of the angular momentum vector. If your object is starting at rest, the instantaneous axis of rotation would be in the direction of dL/dt. Otherwise it would be in the direction of L, which may be changing with time. Note: a torque relative to the center of mass is given by R x F (vector product).

• R.W. Bird I don't think that instantaneous axis of rotation always points in the direction of angular momentum vector. There are a lot of counter-examples to this. – Shivansh J Aug 23 at 18:43