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

= 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)