I'm working my way through "Vector Analysis and Cartesian Tensors by Bourne & Kendall, and I got stuck on problem 3.29.
Wouldn't an anguar velocity of an object from the point of view of a point fixed to it be zero, since the point itself is also experiencing this angualr velocity? I don't want a solution, just a hint at how to approach it.
Problem statement:
The point O lies on Earth's surface at co-latitude $\theta$ (ie. the angle between Earth's diameter through O and the polar axis). Axes Oxyz fixed to the Earth are chosen such that Ox points east, Oy points north and Oz points vertically upwards. Show that, relative to the axes Oxyz , the Earth's angular velocity is $\vec{\omega}=(0, \omega sin(\theta), \omega cos(\theta))$ where $\omega$ is the angular speed of Earth.
Edit: Thanks to @joriki, I managed to solve this bit; but actually, I have trouble with understanding the rest of this question too. I feel like somebody wrote down a question, realised it's too simple and so tried to make it more complicated by describing it in the most convoluted way.
The rest of the problem:
A particle P moves across an ice-rink which may be assumed to be frictionless and is in the xy-plane. The particle is projected from O eastward with initial speed u. If the coordinates of P at a later time t are (x,y,0) show that $\ddot{x}=2\omega\dot{y}cos\theta$ , $\ddot{y}=-2\omega\dot{x}cos\theta$ . Deduce that $\dot{x}^2+\dot{y}^2=u^2$. Find equation of the path of P and deduce that P moves in the xy-plane with constant speed of u along a circular arc of radius $\frac{u}{2\omega cos\theta}$.
I mostly have trouble understanding how this coordinate system is supposed to lay? Does it follow Earth's rotation? Am I still in that coordinate system in this part?
And I feel like I'm supposed to simultaneously ignore and not ignore the curvature of Earth's surface.
This is a very badly formulated exercise. You're right that the correct answer to the question as one would usually interpret it is zero. But apparently they don't mean the angular velocity in a coordinate system with these axes fixed to the Earth, but rather the angular velocity in an inertial frame, expressed along the given axes. It's very misleading to use the word “fixed” for that. A clear statement of the question whose answer is $(0,\omega\sin\theta,\omega\cos\theta)$ would be: