Serway & Jewett — Circular Motion and Other Applications of Newton's Laws: Problem 28

A child’s toy consists of a small wedge that has an acute angle \(\theta\) (Fig. P6.28). The sloping side of the wedge is frictionless, and an object of mass \(m\) on it remains at constant height if the wedge is spun at a certain constant speed. The wedge is spun by rotating, as an axis, a vertical rod that is firmly attached to the wedge at the bottom end. Show that, when the object sits at rest at a point at distance \(L\) up along the wedge, the speed of the object must be \(v = (gL \sin \theta)^{1/2}\).