Acceleration in Ecuador (Yet Another With Linear And Rotational Quantites)

[Pat2666]

I've got another one today that I can't figure out. Here's the problem :

http://img213.imageshack.us/img213/8419/ecuadorearth2dd87dfay2.jpg [Broken]

The Earth has a radius of 6.38 X 106 m and turns on its axis once every 23.9 h.

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a) What is the tangential speed of a person living in Ecuador, a country that lies on the equator?
v = m/sec *
465.91 OK
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b) What is the centripetal acceleration of that person?
acen = m/sec2 *
0.034 OK
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c) At what latitude (i.e. the angle θ in the drawing) is the tangential speed 1/3 that of a person living in Ecuador?
θ =

HELP: The tangential velocity depends on the distance from the axis of rotation. In this case, what is the axis of rotation?
HELP: How can you write the distance to the axis of rotation in terms of the radius of the Earth and the angle?

Okay, so I figured out A & B with no problem, but I've been stuck on C for awhile. I've tried two different methods, but haven't been able to come out with the correct angle. The two HELPs are provided as part of the problem (it's not me saying help :P). I'm a bit confused about making the axis of rotation in terms of the radius of the Earth and the angel. Below is my work for how I attempted to solve C. If something's unreadable please let me know and I'll make it clear :)

http://img213.imageshack.us/img213/1421/workhx4.jpg [Broken]

If I'm not mistaken, w should be the same for any position along "x" right?

 

[Doc Al]

Yes, the angular velocity is the same everywhere. But the distance from the axis is not. First thing to do is figure out the distance from the axis as a function of angle. (That will be the "r" needed in the formula v = wr.) Hint: What's the component of the Earth's radius (the red arrow in the diagram) perpendicular to the axis of rotation for a given angle? (Hint 2: You're using the wrong trig function.)

 

[Pat2666]

Once again thank you!

For some reason I was thinking that the person would be somewhere closer to the Earth on the x-axis, but that would he/she was inside of the Earth now wouldn't it? lol