Calculate Particle Velocity/Accel in Referential w/ Motion & Coriolis Problem

In summary, the problem is to calculate the velocity and acceleration of a particle from a moving referential frame in an inertial reference frame. This referential frame is undergoing both translation and rotation. The corresponding expressions for tangential, centripetal, and Coriolis accelerations need to be identified. The student considers using Galileo's transformations but realizes that they are not applicable in a non-inertial reference frame. They then explore using the Transport theorem and the relation for the derivative of a unit vector to find the desired expressions. With some guidance, they are able to successfully identify the Euler acceleration, Coriolis acceleration, and centrifugal acceleration.
  • #1
fluidistic
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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.


Homework Equations


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


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 [tex]a(t)=\ddot f(t)[/tex], [tex]v(t)=\dot f(t)[/tex] and [tex]r(t)=f(t)[/tex].
But I've no clue about the rotational part. Also big troubles with the translational part. I'd like some guidance.
Thanks.
 
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  • #3
  • #4


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 [itex]\vec r (t)[/itex]), I could show that [itex]\vec v (t)=\vec v _r (t)+ \vec \Omega \times \vec r[/itex]; just like they did.
However I've a problem when it comes to find the acceleration.
If I set their "f" to [itex]\vec v (t)[/itex] then I find [itex]\vec a (t)=\vec a _r (t) + \vec \Omega \times \vec v _r (t)+ \vec \Omega \times \vec \Omega \times \vec r (t)[/itex] instead of [itex]\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)[/itex].
I've no idea what I'm doing wrong.
By the way this problem seems really tough :/
 
  • #5


Use the relation:
[tex]\frac{d}{dt}\hat{\boldsymbol{u}} = \boldsymbol{\Omega \times \hat{u}}[/tex]

I will do the first term for the derivative of the velocity (use the product rule):
[tex]\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}[/tex]

Then you would do the same for the last term in the velocity. You'd take its derivative and use the product rule.
[tex]\frac{d}{dt} \boldsymbol{\Omega \times r}[/tex]
 
Last edited:
  • #6


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


Yep.
 
  • #8


Ok thanks for all. Problem solved. :)
 

1. How do you calculate particle velocity in a referential with motion?

To calculate particle velocity in a referential with motion, you need to use the vector addition formula. This involves adding the velocity of the referential frame to the velocity of the particle in that frame. The resulting vector will be the particle's velocity in the overall referential with motion.

2. What is the Coriolis effect and how does it affect particle velocity?

The Coriolis effect is a phenomenon caused by the rotation of the Earth. It causes objects to appear to curve to the right in the northern hemisphere and to the left in the southern hemisphere. In terms of particle velocity, the Coriolis effect can cause a change in the direction of the particle's motion.

3. Can you explain how to calculate particle acceleration in a referential with motion?

To calculate particle acceleration in a referential with motion, you need to use the vector addition formula for acceleration. This involves adding the acceleration of the referential frame to the acceleration of the particle in that frame. The resulting vector will be the particle's acceleration in the overall referential with motion.

4. What is the difference between particle velocity and acceleration in a referential with motion?

Particle velocity in a referential with motion refers to the speed and direction at which the particle is moving in relation to the moving referential frame. On the other hand, particle acceleration in a referential with motion refers to the rate at which the particle's velocity is changing in relation to the moving referential frame.

5. How can you account for both motion and the Coriolis effect when calculating particle velocity and acceleration?

To account for both motion and the Coriolis effect when calculating particle velocity and acceleration, you need to use the vector addition formula for both velocity and acceleration. This involves adding the respective vectors for motion and the Coriolis effect to the particle's velocity and acceleration in the stationary referential frame. This will result in the particle's velocity and acceleration in the overall referential with motion.

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