Classical Mechanics: Coriolis Effect Problem

[Eyedbump]

Homework Statement

A bird of mass 2 kg is flying at 10 m/s in latitude of 60° N, heading due East. Find the horizontal and vertical components of the Coriolis force acting on it.

Homework Equations

The Coriolis Force, F = 2mw∧v. Where ∧ shows the cross product between angular frequency vector, w, and change in the position vector, v.

Θ will be the co-latitude -- that is, 90°- 60° = 30°.

The Attempt at a Solution

I started by deciding that my coordinates would be oriented so that x points East, y points North, and z points straight up (away from the earth). Thus, I believe, w = {wcosΘ, 0, 0} since the bird flies only East.

So taking the Cross product with v = {x' , y' , z'} (where ' indicates the change in position), I receive the following vector {0 , -z'cosΘ , y'cosθ}. Now, I've shown the product vector without the coefficients, because my confusion arises at the presence of the y' and z's. Exactly what am I to do about them?

It's one of those problems where I can't tell if I'm missing something terribly basic, or having been working under a more general misapprehension. I'd very much appreciate any help!

p.s. This is my first post in the forum, and so I'm sure I've broken a plethora of the rules/etiquettes for which you must forgive me.

p.p.s. This is not a homework problem, just a kind of review (which makes the fact that I'm struggling with it so much more embarrassing), so don't feel ashamed at helping me cheat!

 

[ehild]

Welcome to PF!

Homework Statement

A bird of mass 2 kg is flying at 10 m/s in latitude of 60° N, heading due East. Find the horizontal and vertical components of the Coriolis force acting on it.

Homework Equations

The Coriolis Force, F = 2mw∧v. Where ∧ shows the cross product between angular frequency vector, w, and change in the position vector, v.

You miss a minus sign. The Coriolis force is F = -2mw∧v.

Θ will be the co-latitude -- that is, 90°- 60° = 30°.

The Attempt at a Solution

I started by deciding that my coordinates would be oriented so that x points East, y points North, and z points straight up (away from the earth). Thus, I believe, w = {wcosΘ, 0, 0} since the bird flies only East.

The angular velocity is a vector parallel to the axis of rotation of Earth and pointing upward. In your coordinate system it has both y and z components, and zero x component.

So taking the Cross product with v = {x' , y' , z'} (where ' indicates the change in position), I receive the following vector {0 , -z'cosΘ , y'cosθ}. Now, I've shown the product vector without the coefficients, because my confusion arises at the presence of the y' and z's. Exactly what am I to do about them?

The velocity vector is (10, 0,0) as it has only East (x) component.

 

[Eyedbump]

Welcome to PF!You miss a minus sign. The Coriolis force is F = -2mw∧v.
The angular velocity is a vector parallel to the axis of rotation of Earth and pointing upward. In your coordinate system it has both y and z components, and zero x component.
The velocity vector is (10, 0,0) as it has only East (x) component.

Oh my god. Thank you so much!

 

[ehild]

You are welcome. :)

 

[HalfLostAndSearching]

Hello,

Thank you for reaching out for help on this problem. I can provide a response to your question.

Firstly, it's important to note that the Coriolis force is a fictitious force that arises due to the rotation of the Earth. This force only appears to act on objects in motion and is not a real force like gravity or electromagnetism.

Now, let's look at the given problem. We have a bird flying at a constant velocity of 10 m/s due East in a latitude of 60°N. The Coriolis force acts perpendicular to the direction of motion, so in this case, it would act in the North-South direction.

The equation you have provided for the Coriolis force is correct, but we need to take into account the direction of the force. Since the bird is flying due East, the velocity vector would be in the x-direction. Therefore, the cross product between the angular frequency vector, w, and the change in position vector, v, would give us a force vector in the y-direction.

The angular frequency vector, w, is given by w = {0, 0, w}, where w is the angular frequency of the Earth's rotation. In this case, since the bird is flying at a latitude of 60°N, we can use the formula w = 2ΩsinΘ, where Ω is the angular velocity of the Earth's rotation and Θ is the latitude. Thus, w = 2Ωsin60° = √3Ω.

Now, let's look at the change in position vector, v. Since the bird is flying at a constant velocity of 10 m/s due East, the change in position vector would be {10t, 0, 0}, where t is the time.

Taking the cross product of w and v, we get the Coriolis force vector, F = {0, 0, 2√3Ωt}.

To find the horizontal and vertical components of this force, we need to project this vector onto the x-y plane and the z-direction, respectively.

The horizontal component would be Fx = 0 and the vertical component would be Fz = 2√3Ωt.

Therefore, the horizontal and vertical components of the Coriolis force acting on the bird would be zero and 2