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Acceleration in a rotating frame

  1. May 11, 2005 #1
    I'm getting confused by this. I have a handout from a lecture that has a derivation that ends with

    "[tex]\vec{a} = \vec{a'} + 2\vec{\omega} \times \vec{v'} + \vec{\omega} \times (\vec{\omega} \times \vec{r})[/tex]

    Multiplying through by mass, m

    [tex]m\vec{a} = \vec{F_{ext}} = m\vec{a'} + 2m\vec{\omega} \times \vec{v'} + m\vec{\omega} \times (\vec{\omega} \times \vec{r})[/tex]

    We preserve Newton II in rotating frame by writing [tex]\vec{F'_{net}} = m\vec{a'}[/tex] where [tex]\vec{F'_{net}}[/tex] is the net force measured by observer in rotating frame.

    ie. [tex]\vec{F'_{net}} = \vec{F_{ext}} - 2m(\vec{\omega} \times \vec{v'}) - m[\vec{\omega} \times (\vec{\omega} \times \vec{r})][/tex]"

    It's really the last line that's confusing me. The expressions for the Coriolis and centrifugal forces are

    [tex]\vec{F_{Cor}} = -2m(\vec{\omega} \times \vec{v'})[/tex] and [tex]\vec{F_{cent}} = -m\vec{\omega} \times (\vec{\omega} \times \vec{r})[/tex], so why isn't it

    [tex]\vec{F'_{net}} - \vec{F_{Cor}} - \vec{F_{cent}} = \vec{F_{ext}}[/tex]?
     
  2. jcsd
  3. May 11, 2005 #2

    dextercioby

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    Hold on a second,who's [itex] \vec{F}_{ext} [/itex]...?

    Daniel.
     
  4. May 11, 2005 #3
    "Sum of real forces (electrical, magnetic, gravitational, etc); only these forces are observed in stationary frame".

    All I'm getting confused about is the signs of those forces.
     
  5. May 11, 2005 #4

    dextercioby

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    Why shouldn't the NET force be the sum of all external forces...?Afer all,both Coriolis & centrifugal are inertial forces,they're not external forces.

    Daniel.
     
  6. May 11, 2005 #5
    Grr, I know that, but:

    [tex]m\vec{a} = \vec{F_{ext}} = m\vec{a'} + 2m\vec{\omega} \times \vec{v'} + m\vec{\omega} \times (\vec{\omega} \times \vec{r})[/tex]

    ie. [tex]\vec{F'_{net}} = \vec{F_{ext}} - 2m(\vec{\omega} \times \vec{v'}) - m[\vec{\omega} \times (\vec{\omega} \times \vec{r})][/tex]"

    The first and second lines aren't the same. If [tex]\vec{F'_{net}} = m\vec{a'}[/tex], then the first line is [tex]m\vec{a} = \vec{F_{ext}} = \vec{F'_{net}} - \vec{F_{Cor}} - \vec{F_{cent}}[/tex] :confused:.
     
    Last edited: May 11, 2005
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