Total Force on Body in Rotating Frame of Reference

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $B$ be a body in a rotating frame of reference $\FF$.

The total force $\mathbf {F'}$ on $B$ is given by:

\(\ds \mathbf {F'}\) \(=\) \(\ds \mathbf F - m \dfrac {\d \bsomega} {\d t} \times \mathbf {r'} - 2 m \bsomega \times \mathbf {v'} - m \bsomega \times \paren {\bsomega \times \mathbf {r'} }\)
\(\ds \) \(=\) \(\ds m \mathbf {a'}\)

where:

$\mathbf F$ is the vector sum of the forces acting on $B$
$\bsomega$ is the angular velocity of $\FF$ relative to an inertial frame
$\mathbf {r'}$ is the position vector of $B$ relative to $\FF$
$\mathbf {v'}$ is the velocity of $B$ relative to $\FF$
$\mathbf {a'}$ is the acceleration of $B$ relative to $\FF$.


The inertial forces as they are perceived in the rotating frame of reference $\FF$ are:

\(\ds \) \(\) \(\ds -m \dfrac {\d \bsomega} {\d t} \times \mathbf {r'}\) the Euler force
\(\ds \) \(\) \(\ds -2 m \bsomega \times \mathbf {v'}\) the Coriolis force
\(\ds \) \(\) \(\ds -m \bsomega \times \paren {\bsomega \times \mathbf {r'} }\) the centrifugal force


Proof


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Total_Force_on_Body_in_Rotating_Frame_of_Reference&oldid=663945"