= Solution
<Conservation of energy> states that the total <energy> of an isolated system is constant. In a mechanical problem with conservative <forces> and no dissipation, it gives $K+U=\text{constant}$. It often determines a <speed> or turning height directly, without solving the full time-dependent <equation of motion>. With external work or dissipation, use the corresponding <energy balance> instead.
After a drop leaves the wheel, neglecting air resistance, the conservative <force> of the <Newtonian gravitational field> acts. Put the origin at the wheel axis and choose positive horizontal displacement to the right. In the printed diagram $\theta$ is measured upward from the leftward horizontal radius, and rotation is clockwise. The release position and <velocity> are therefore
$$
(x_0,y_0)=(-r\cos\theta,r\sin\theta),\qquad (v_{x0},v_{y0})=(r\omega\sin\theta,r\omega\cos\theta).
$$
For upward release, $\cos\theta>0$. These conventions fix the signs used below.
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