Angular velocity in body-fixed frame and space-fixed frame

Question

When we solve for a free symmetric top we find that in body fixed frame, the angular velocity precesses. My confusion is regarding the calculation of omega in body frame. When i am in the body fixed frame, shouldn't the angular velocities be zero, because everything is fixed in this frame?

Answer 1

The angular velocity of some object with respect to an inertial frame can be expressed in any frame. Oftentimes it is expressed in a body-fixed frame. This practice dates back to Leonhard Euler. It turns out to be quite practical because that is the frame in which angular velocity sensors such as gyroscopes sense angular velocity.

Answer 2

I agree this is confusing. Torque and angular momentum are free vectors and are therefore the same in both the fixed space and rotating body axes; the time derivative of a free vector (e.g., the angular momentum) is not the same and the magnitudes of the components of the free vectors along the axes are not the same. The Euler equations for the rotation of a rigid body utilize rotating body coordinates chosen to be principal axes, because the components of the inertia tensor are constant relative to the body axes. The Euler equations express the time derivative of angular momentum in the fixed space coordinates in terms of the rotating body coordinates and relate that to the external torque in the fixed space coordinates. This is not viewing the motion in the non-inertial rotating body coordinates; it is expressing the components of the angular momentum and torque along the rotating body axes. (If you express the motion in the rotating frame you need to consider fictitious torques due to the fictitious forces that arise in a non-inertial- accelerating- frame.) A good physics mechanics text should help understand all this; for example, Symon Mechanics or Goldstein Classical Mechanics.

The angular velocity is the angular velocity of the rotating body axes with respect to the fixed space axes. Remember, you are expressing motion in terms of the body axes coordinates, you are not moving with the body axes.

Answer 3

This is a common misconception. The equations of motion always need to be stated on an inertial reference frame (so attached to the ground, or with constant velocity). But we have a choice on the orientation of the coordinate system.

A better term is choosing the basis vectors of the equations of motion.

So body rotational velocity (as well as any other vectors) can be expressed in either body aligned (instantaneously) directions, or ground aligned (fixed) directions.

In the first case the MMOI tensor is fixed, but integrating over time is more complex because the orientation changes.

In the second case the MMOI tensor is changing, but integrating over time is easier.

So when talking about the body frame, we are really talking about an inertial frame that is aligned with the body at some instant, and not a body riding frame.

Answer 4

Consider you are sitting on a constant rotating disc with high walls, so you can't see the surrounding IS. Surely all points of the disc are rotating with you, so $\omega_{disc}=0$ for your reference system. But this is not an inertial system (IS). So, if you want to calculate any dynamics in your frame, you have to correct your Newtonian equation of motion by (in this case two) fictions forces, that depend from the $\omega_{IS}$ seen in IS. This $\omega_{IS}$ is so to say a kind of field-strength in your own reference system. You can measure it by examination of forces or paths of particles and it is not necessary to see the IS-"world" around you. For your spinning top, the "$\omega_{disc}(t)$-field" would additionally be time-dependent due to the superposition of different rotations.