1
$\begingroup$

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?

$\endgroup$

1 Answer 1

1
$\begingroup$

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 $a = \frac{F_{lift}}{m} = \frac{c_f\Omega}{m}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.