Coriolis Effect: Calculate Acceleration & Direction

[Chowie]

Homework Statement

On its way to Paris the Eurostar train is traveling due South at 300 km/h
at a point with latitude 49°. Assume the Earth is a perfect sphere of radius,
R_E, that rotates around its axis (North/South pole) once every 23 hrs 56
min.
(a) Calculate the magnitude of the acceleration of the train due to the
Coriolis force. Note that latitude is defined as the angle $\phi$ with respect
to the North-South pole axis where $\phi$ = 0 at the equator and $\phi$ = 90° at the North pole.
(b) What direction is this acceleration in?

Homework Equations

Acceleration due to coriolis effect:

$\vec{a}$_co = 2$\vec{v}$×$\vec{ω}$

where a_co is the accerlation due to coriolis effect, $\vec{v}$ is the velocity of the object in question and $\vec{ω}$ the angular frequency at that latitude.

The Attempt at a Solution

I know this question has something to do with 2 separate reference frames, one inertial and one not, however for some reason I seemed to have ignored that completely and just attempted it this way:

First of all I calculated the angular frequency at 49 degrees latitude this way:

angular frequency at equator: $\omega = 2\pi/\tau$
angular frequency at 49° latitude = $\omega = (2\pi/\tau) cos(49°)$
$\tau = 23+14/15$ hours

Then I moved on to converting the velocity of the train into spherical co-ordinates. I did this via ratios rather than standard conversions because well, I don't really know it just seemed easier, it's probably where I've gone wrong, I've drawn a diagram on my sheet of paper but hopefully it'll make sense without it.

Ok so the train is traveling due south on the surface of the earth, with a velocity of 300 km/h, over a total distance of half the circumference of the Earth (as the train is starting from a lower latitude it isn't actually covering this distance but hopefully that is irrelevant), what I'm looking for is $\partial\phi/\partial t$ and I calculated that:

$\partial v/ \partial t / 0.5 Ce= (\partial \phi / \partial t) / \pi$

Where Ce = circumference of earth.

I then solved for $\partial\phi/\partial t$ and got this equal to 0.047 rad / hour in the [itex]$\widehat{\phi}$ direction.

I then did the cross product of my angular frequency of the Earth at this point and the angular velocity multiplied by 2 and my result came out in the $\widehat{r}$ direction, and was negative so it is going into the planet. I know the drift should be to the left so I'm real confused.

I'm pretty sure I've approached it all wrong so I'd appreciate any help. Thanks.

 

[ehild]

See:
http://en.wikipedia.org/wiki/Coriolis_effect. Scroll down to find "Rotating Sphere" .ehild

 

[ehild]

I can not follow your work. The Coriolis force acts in a rotating frame of reference. Choose a coordinate system on the surface of the Earth, which moves together with the rotating Earth. Wikipedia explains it quite nicely: The axis of the local system of coordinates point to East, North, and Up with unit vectors $\hat{e}$, $\hat{n}$ and $\hat{u}$.The vector of angular velocity of the Earth is the same everywhere, but its components in a local frame depend on latitude: $\vec{Ω}$=cosφ$\hat{n}$+sinφ$\hat{u}$. The velocity of the train going to South is $\vec{v}$=-v$\hat{n}$.

Calculate the vector product $\vec{v}$x$\vec{Ω}$. What are its non-zero components?. What is the direction of the product?

ehild

 

[Chowie]

Yeah I was approaching this all wrong, I looked at the method on wikipedia a few days ago and it worked out fine, thanks for the input.

 

[ehild]

Hi Chowie,

Next time send a post, please, if you solved the problem. I thought you gave up all hope and left Physicsforums forever, not getting a proper help.

ehild