Calculating Angle Between Plumb Line & Radius Vector Over North Pole

AI Thread Summary
The discussion revolves around calculating the angle between a plumb line in a jet plane flying due south over the North Pole and the radius vector from the Earth's center to the plane. Participants express confusion about the definition of the radius vector and the assumption that the Earth's angular velocity of 2π radians in 24 hours is merely an approximation rather than an exact value. Clarification is provided that this approximation relates to the difference between sidereal and solar days. The problem involves understanding the plane's path relative to the Earth's rotation and suggests that finding three points along the path could help determine the curve. Overall, the discussion emphasizes the complexities of the geometry involved in this scenario.
thesaruman
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Homework Statement



A jet plane flies due south over the north pole with a constant speed of 500 mph. Determine the angle between a plumb line hanging freely in the plane and the radius vector from the center of the Earth to the plane above the north pole.
Hint, assume that the Earth's angular velocity is 2pi radians in 24 hr, which is a good approximation. Why?

Homework Equations


The Attempt at a Solution



Well, I just can't figure out what radius vector is this... I mean, which plane is this? And I didn't understand why the hypothesis of an Earth's angular velocity of 2pi radians in 24hr is an approximation. I thought that this could be considered an exact parameter.
 
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thesaruman said:

Homework Statement



A jet plane flies due south over the north pole with a constant speed of 500 mph. Determine the angle between a plumb line hanging freely in the plane and the radius vector from the center of the Earth to the plane above the north pole.
Hint, assume that the Earth's angular velocity is 2pi radians in 24 hr, which is a good approximation. Why?

Homework Equations



dx = v dt is one and v = \omega v dt \sin ( \theta ) is the other.

The Attempt at a Solution



Well, I just can't figure out what radius vector is this... I mean, which plane is this? And I didn't understand why the hypothesis of an Earth's angular velocity of 2pi radians in 24hr is an approximation. I thought that this could be considered an exact parameter.

Considering that this plan is EXACTLY above the North pole, and that in the initial instant of time the jet plane is flying through this radius vector, the answer would depend of a time interval. What I could do?
I mean, the plane would be a distance dx = v dt \hat{\mathbf{x}} in \hat{\mathbf{x}} direction and simultaneously, the Earth would have turned an angle equal to \frac{d\omega}{dt} in \hat{\mathbf{\phi}} direction. I just can't eliminate the time from the solution.
 
First, just cleaning up the LaTex stuff so it displays properly:
... the plane would be a distance
dx = v dt \hat{\mathbf{x}} \ \mbox{in} \ \hat{\mathbf{x}} \ \mbox{direction}
and simultaneously, the Earth would have turned an angle equal to
\frac{d\omega}{dt} \ \mbox{in} \ \hat{\mathbf{\phi}} \ \mbox{direction.}

It looks like they want you to find the curve in the plane's path, which is straight line only relative to the rotating Earth. I'm not sure off the top of my head how to do that, but if you can find 3 points along the path it should be possible to fit a circle to them.

I didn't understand why the hypothesis of an Earth's angular velocity of 2pi radians in 24hr is an approximation. I thought that this could be considered an exact parameter.

It has to do with a sidereal vs. solar day.
 
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