# Reference frame, referential with motion, describe the motion of a particle/Coriolis

1. Apr 20, 2010

### fluidistic

Problem:
Calculate the velocity and acceleration from an inertial reference frame of a particle whose motion functions (in Cartesian's coordinates) are known from a moving referential. The motion of such a referential is in accelerated translation and rotation with respect to the inertial one. Identify the corresponding expression of tangential, centripetal and Coriolis's accelerations.

2. Relevant equations
Galileo's transformations... hmm not sure. I don't think so since I'm dealing with a non inertial reference frame.

3. The attempt at a solution
I'd like some guidance. I'm thinking of starting writing the motion of the particle the referential sees but I have a big confusion when it comes to the rotational part (is it a spin and an orbital motion?).
My other idea is to start to write down a similar relation to Galilean's transformation.
The translation from one frame to another involves an acceleration. I call it $$a(t)=\ddot f(t)$$, $$v(t)=\dot f(t)$$ and $$r(t)=f(t)$$.
But I've no clue about the rotational part. Also big troubles with the translational part. I'd like some guidance.
Thanks.

2. Apr 21, 2010

### nickjer

3. Apr 21, 2010

### fluidistic

Re: Reference frame, referential with motion, describe the motion of a particle/Corio

Thanks, very relevant. I will take time to read it and use it. I'll come back here if I need to.

4. Jul 12, 2011

### fluidistic

Re: Reference frame, referential with motion, describe the motion of a particle/Corio

I know it's been more than 1 year since I've posted but here I am.
In Wikipedia's article and using what they call the Transport theorem (replacing their "f" by $\vec r (t)$), I could show that $\vec v (t)=\vec v _r (t)+ \vec \Omega \times \vec r$; just like they did.
However I've a problem when it comes to find the acceleration.
If I set their "f" to $\vec v (t)$ then I find $\vec a (t)=\vec a _r (t) + \vec \Omega \times \vec v _r (t)+ \vec \Omega \times \vec \Omega \times \vec r (t)$ instead of $\vec a (t)=\vec a _r (t) +2 \vec \Omega \times \vec v _r (t)+ \vec \Omega \times (\vec \Omega \times \vec r )- \frac{d \vec \Omega }{dt} \times \vec r (t)$.
I've no idea what I'm doing wrong.
By the way this problem seems really tough :/

5. Jul 12, 2011

### nickjer

Re: Reference frame, referential with motion, describe the motion of a particle/Corio

Use the relation:
$$\frac{d}{dt}\hat{\boldsymbol{u}} = \boldsymbol{\Omega \times \hat{u}}$$

I will do the first term for the derivative of the velocity (use the product rule):
$$\frac{d}{dt}\boldsymbol{v_r} = \frac{d}{dt}v(t)\hat{\boldsymbol{r}} = \frac{dv(t)}{dt}\hat{\boldsymbol{r}} + v(t) \frac{d\hat{\boldsymbol{r}}}{dt} = a(t)\hat{\boldsymbol{r}} + v(t) \boldsymbol{\Omega \times \hat{r}} = \boldsymbol{a_r} + \boldsymbol{\Omega \times v_r}$$

Then you would do the same for the last term in the velocity. You'd take its derivative and use the product rule.
$$\frac{d}{dt} \boldsymbol{\Omega \times r}$$

Last edited: Jul 12, 2011
6. Jul 14, 2011

### fluidistic

Re: Reference frame, referential with motion, describe the motion of a particle/Corio

Thank you very much, I could demonstrate the formula in wikipedia thanks to you.
I could identify Euler acceleration, Coriolis acceleration and the centrifugal acceleration.
However in the question, they ask for the "tangential acceleration", do they mean the Euler acceleration by this term?

7. Jul 14, 2011

### nickjer

Re: Reference frame, referential with motion, describe the motion of a particle/Corio

Yep.

8. Jul 14, 2011

### fluidistic

Re: Reference frame, referential with motion, describe the motion of a particle/Corio

Ok thanks for all. Problem solved. :)