Monday, February 4, 2019

newtonian mechanics - Time derivative of vector in rotating frame with angular velocity about a rotating axis


In general, I know that if you have a vector $\vec{F}$ in a rotating frame, and the frame has an angular velocity $\vec{\Omega}$ that the time derivative of $\vec{F}$ in a fixed frame would be $$\frac{d\vec{F}}{dt}=\left(\frac{d\vec{F}}{dt}\right)_r+\vec{\Omega}\times\vec{F}.$$


However, I'm confused how or if this would change if there are multiple angular velocities attached to a rotating axis. Let's say our rotating frame is as below. Angular Velocities


This angular velocity $\vec{\Omega_{z'}}$ has its own angular velocity $\vec{\Omega_y}$. My original thoughts are to simply combine the angular velocities into a single vector $\vec{\Omega_T}=\vec{\Omega_y}+\vec{\Omega_{z'}}$, but since the axis $z'$ is moving I'm not sure if it's that simple.



Answer



As was mentioned in the comments, there is only one angular velocity $\vec{\Omega}_T=\Omega_y\hat{y}+\Omega_{z'}\hat{z'}$. This is confirmed here from some MIT lecture notes. It seems my intuition was correct. EDIT: If you want to use this to find the velocity of a vector, you need to cast this into the global frame first.


No comments:

Post a Comment

classical mechanics - Moment of a force about a given axis (Torque) - Scalar or vectorial?

I am studying Statics and saw that: The moment of a force about a given axis (or Torque) is defined by the equation: $M_X = (\vec r \times \...