Deriving Ball Movement on Northern Hemisphere with Rotating Reference Frame

[physicschick21]

I have a ball of mass m that is situated on horizontal plane on the northern Hampshire. I am asked to show that the ball is moving, clockwise, in a manner of

r = v / ( 2Ω*sin(λ) )

where v is the ball's velocity, Ω is Earth's angular velocity, and λ is the terrestrial latitude

So here's what I tried doing:

I know that the Earth's angular velocity is given by Ω = Ω (cosλ,0,sinλ)
In the Northern Hemisphere thus it is given by: Ωsinλ

I think that I need somehow to add Coriolis Force but I'm not sure where or how to start the equation of motion.

Many thanks

 

[timetraveller123]

Forum rules is that you have to show work
so i will just provide the equation for coriolis force
$F_{coriolis} = -2 m \vec \omega \times \vec v$
where omega is angular velocity of Earth and v is the velocity observed in the rotating frame

think about which way of rotation can provide the necessary centripetal force

 

[physicschick21]

Forum rules is that you have to show work
so i will just provide the equation for coriolis force
$F_{coriolis} = -2 m \vec \omega \times \vec v$
where omega is angular velocity of Earth and v is the velocity observed in the rotating frame

think about which way of rotation can provide the necessary centripetal force

Thank you. So this is where I got so far: the equation of motion is given, in this case, by:

0 = m*Ω x (Ω x r) - 2m * (Ω x v)

I could eliminate the mass from both sides.
and I also know that the motion is perpeniduclar to the rotation axis, so the coriolis force could be just: 2*m*w*v

so I'm left with:

0 = Ω x (Ω x r) - 2Ωsinλ*v

so I only need to calculate the left hand term.. I know what Ω is: Ω = Ω (cosλ,0,sinλ)
but how do I write r? (x,y,z)?

Thanks a bunch for the help!

 

[timetraveller123]

I got so far: the equation of motion is given, in this case, by:

since the question only says show it moves in clockwise i don't think you might actually need any equation

0 = m*Ω x (Ω x r) - 2m * (Ω x v)

i don't quite get what is happening here why are you setting it to zero

the second sort of circle there is meant to be motion of ball and is in the tangent plane
if we assume Earth is perfectly spherical then there are two forces on the ball the observed centrifugal force shown in the picture and Coriolis force
this coriois force actually has two components
it has
$2m \Omega sin \lambda$
in the tangent plane (this is the one you need)
the other one is
$2m \Omega cos \lambda \sin \phi$
where $\phi$ is the angle of rotation of the mass in its own circle but this term points perpendicular to tangent plane and the normal force from the Earth can adjust to accommodate it
so actually the only two forces in the tangent plane are
$-2m \Omega v sin \lambda$
and
$m \Omega \Omega R sin \lambda$

try to take it from here
another way would be that
the Earth is actually not perfectly spherical which means the centrifugal force can be ignored i think as the normal force can account for that also then you only have to concern with coriolis force either way they will rotate clockwise in north

edit :
i just realized that if you assume Earth is perfectly spherical there is no way of definitively saying it rotates clockwise unless you are given V as you don't know which term outweighs

unless 2V is greater than $\Omega R$ = 1000mph we can't use the first method so i think we have to assume Earth is not spherical and ignore the centrifugal force altogether

edit 2
i just realized you were asked to derive the equation for radius so you definitely need to do equations and based on the form of the equation you definitely need to ignore centrifugal term

The more experienced pf members help me verify thanks

 

[timetraveller123]

let me clear up my last post
the four forces on the ball are
centrifugal normal gravitational and Coriolis
by the construction of shape of Earth all but the Coriolis force sum to zero
so in essence the only force you need to concern yourself with is the component of coriolis along the tangent plane and the condition is that this need to provide the necessary centripetal force

 

[physicschick21]

Ok! I think I understand what you wrote. I'll try to derive the equation for r.. hopefully this works out. You are superb!

 

[timetraveller123]

Oh and btw welcome to physics forum

 

[physicschick21]

Oh and btw welcome to physics forum

Thank you :)
You're very kind!

I'm still stuck and not getting this. I'll try to read some more online and maybe I'll understand what I'm missing and get the exact equation they're looking for

 

[physicschick21]

Thank you! I understood everything now! :D