Angular velocity - spherical coordinates

In summary, the conversation discussed how to express angular velocity in spherical coordinates and how to specify the direction of the angular velocity vector. The magnitude of the Coriolis force was also mentioned and it was determined that it is independent of the direction of motion of the particle. The most common way to express spherical coordinates was also mentioned. Finally, the questioner requested the angular velocity of the Earth in spherical coordinates and received clarification on the unit vectors and cross product involved.
  • #1
emptymaximum
110
0
how do you express angular velocity in spherical coordinates?
like the Earth rotates with constant speed, so the direction of the angular velocity vector is out the north pole.
if it was spherical coordinates , how do you specify that direction?
i know that z = r cos theta so
[tex] \hat{k} = r cos \theta \hat{e_{\theta}} [/tex] ?
the problem is:
for a particle moving in a horizontal plane on the surface of the Earth show that the magnitude of the coriolis force is independent of the direction of motion of the particle.
so the question i asked isn't the HW problem and a direct answer would be much appreciated.
also, this is sort of a dumb question, but does the [itex] \theta , \phi [/itex] plane count as a plane? if it doesn't, why wouldn't it?
 
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  • #2
z hat, what's a zhat?

[tex] z= \rho cos(\phi) [/tex]
 
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  • #3
Generally, you would express any vector in spherical coordinates in the form of

[tex]\vec v = v_r \hat e_r + v_{\theta} \hat e_{\theta} + v_{\phi} \hat e_{\phi}[/tex]
 
  • #4
i know how a velocity is expressed in spherical co-ordinates.
i just don't know how to say the direction of the angular velocity vector.
magnitude of the coriolis force =
[tex] F_{cor} = | 2 m ( \vec{\omega} \times \vec{v'} ) | [/tex]
where [tex] \vec{v'} [/tex] is the velocity in the non-inertial frame.
wht's rho? am i using the wrong notation for this too?
[tex] ( \hat{e_r} , \hat{e_{\theta}} , \hat{e_{\phi}} ) [/tex] is i think the most common way to express it, but it's not so type friendly. I generally use:
[tex] ( \hat{r} , \hat{\theta} , \hat{\phi} ) [/tex] for spherical coordinates, but that's besides the point.
now that the confusion to nomenclature has been cleared up:

i need to know the angular velocity of the Earth in spherical coordinates please.
is it just [tex] \omega \hat{k} = \omega r cos \theta \hat{e_{\theta}} [/tex]
 
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  • #5
my rho is your r.
 
  • #6
from your post the #2 i am inferring that i am correct in my assumption that

[tex] \omega \hat{k} = \omega r cos \theta \hat{e_{\theta}} [/tex]
 
  • #7
For the Coriolis force, [itex]\omega[/itex] is in the [itex]\hat z[/itex] direction. It may be helpful to write that as [itex]\hat z = -\sin \theta \hat e_{\theta} + \cos \theta \hat e_r[/itex]. Then if you write the velocity using spherical unit vectors you can evaluate the cross product noting that [itex]\hat e_r \times \hat e_{\theta} = \hat e_{\phi}[/itex] etc.
 
  • #8
thank you.
 

What is angular velocity in spherical coordinates?

Angular velocity in spherical coordinates refers to the rate of change of the angular position of an object as it moves along a spherical path. It is commonly represented by the Greek letter omega (ω) and is measured in radians per unit time.

How is angular velocity calculated in spherical coordinates?

In spherical coordinates, angular velocity is calculated by dividing the change in angular position by the change in time. This can be represented mathematically as ω = Δθ / Δt, where ω is angular velocity, Δθ is the change in angular position, and Δt is the change in time.

What is the relationship between linear and angular velocity in spherical coordinates?

In spherical coordinates, linear and angular velocity are related by the radius of the spherical path. As the radius increases, the same angular velocity will result in a higher linear velocity, and vice versa. This relationship can be expressed as v = ωr, where v is linear velocity, ω is angular velocity, and r is the radius.

Can angular velocity change in spherical coordinates?

Yes, angular velocity can change in spherical coordinates if the object's angular position or speed changes. This can occur if the object speeds up, slows down, or changes direction along the spherical path.

How is angular velocity represented graphically in spherical coordinates?

In spherical coordinates, angular velocity is typically represented using a vector diagram, with the magnitude of the vector representing the angular velocity and the direction of the vector indicating the axis of rotation.

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