A child's toy consists of a small wedge that has an acute...

A child's toy consists of a small wedge that has an acute angle $\theta$ (Fig. $\mathrm{P} 6.28$ ). 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 $v=(g L \sin \theta)^{1 / 2}$

Key Concepts

Centripetal Force and Circular Motion

When an object moves in a circular path, it experiences an inward (centripetal) acceleration that requires a net inward force. This force is typically expressed as m*v^2/r, where m is the mass, v is the tangential speed, and r is the radius of the circular path. In problems where rotational motion is involved, identifying how the forces provide this centripetal force is a key step in understanding the dynamics of the system.

Force Decomposition on an Inclined Plane

On an inclined plane, gravitational force is resolved into components parallel and perpendicular to the surface. The parallel component, mg*sin(?), influences motion along the plane, while the perpendicular component, mg*cos(?), is countered by the normal reaction from the surface. This method of resolving forces is fundamental to analyzing situations where frictionless or frictional inclined surfaces are involved.

Equilibrium in Non-Inertial Reference Frames

When an object is in a rotating frame of reference, it experiences additional apparent forces, such as the centrifugal force, which act outward from the axis of rotation. For an object to remain at rest in such a non-inertial frame, the net force must be zero. Establishing equilibrium by balancing the real forces, like components of gravity, with these apparent forces is crucial in understanding the conditions for steady-state motion in rotating systems.

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