Wikipedia:Reference desk/Archives/Mathematics/2019 January 8

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January 8

Angular accleration definition

I have seen, in an exam question and associated solution, the following. $\mathbf {\alpha } ={\dot {\mathbf {\omega } }}+\mathbf {\Omega } \times \mathbf {\omega }$, where $\mathbf {\alpha }$ is the angular acceleration of a body in a certain (not body-fixed) coordinate system, $\omega$ is, I thought, the angular velocity of the body, and $\mathbf {\Omega }$ is the angular velocity of the frame in which the body's centre of rotation is fixed. I have two questions.

1. Can anyone explain/direct me to an explanation of what is going on, and what the definitions in use are, precisely? I'm aware that slightly different definitions and/or notations are widespread.
2. In the aforementioned frame where the body's centre of rotation is fixed, its axis of rotation is not. Specifically: the body is rotating about an axis A, that axis is rotating about another axis B, and that axis is rotating about an axis C. Yet, I see but a single cross product. Where am I going wrong?--Leon (talk) 12:48, 8 January 2019 (UTC)

The formula is plainly wrong. Dimensions do not add up. Ruslik_Zero 18:05, 8 January 2019 (UTC)

Which is my fault: it should be $\mathbf {\alpha } ={\dot {\mathbf {\omega } }}+\mathbf {\Omega } \times \mathbf {\omega }$.--Leon (talk) 19:15, 8 January 2019 (UTC)

I assume that $\mathbf {\Omega }$ is the vector sum of the separate axial rotations? Dbfirs 22:09, 9 January 2019 (UTC)

No, $\mathbf {\omega }$ is. $\mathbf {\Omega }$ is the only axis of rotation to go through the origin, and the other two axes about which I am calculating the rotation do not go through the origin.

I can try and provide some sort of schematic if that might help.--Leon (talk) 09:14, 10 January 2019 (UTC)

The relation that you wrote above is known as Basic Kinematic Equation in mechanics. It connects the rate of change of a (pseudo)vector in a rotating and a non-rotating reference frames. Ruslik_Zero 18:22, 11 January 2019 (UTC)