Angular velocity and velocity of CM indipendence in rigid body motion

Question

In the most general case, in rigid body motion the linear velocity of the center of mass $v_{cm}$ and the angular velocity of the rigid body $\Omega$ are not related with each other.

Which condition must be satisfied in order to have $v_{cm}$ and $\Omega$ not indipendent one from the other?

Answer 1

Well, my reasoning may be a bit circular, but I'd say that for the linear and angular velocity not to be independent, the one needs to be a function of the other: $v_{cm}=v_{cm}(\Omega)$. I can only think of cases where the relation would be directly proportional. Two examples:

• A ball or cylinder rolling over a surface without slipping: in this case $v_{cm} = \Omega R$, with $R$ the radius of the ball/cylinder.
• An airplane propeller: $v_{cm}(t)=\frac{c_f}{m}\int_{t_0}^{t}\Omega t$, where $c_f$ is some positive constant, $m$ is the mass of the propeller and $t$ is the time.

For the latter I'm assuming that a constant angular velocity $\Omega$ of the propeller produces a proportional lift force $F_{lift} = c_f \Omega$ which will constantly accelerate the propeller by $a = \frac{F_{lift}}{m} = \frac{c_f\Omega}{m}$