Via jkwiens.com, here’s an interesting problem:
A bead slides along a smooth wire bent in the shape of a parabola, . The bead rotates in a circle, of radius , when the wire is rotating about its vertical symmetry axis with angular velocity . Find the constant .
This reminds me of the Classical Mechanics problems I used to enjoy when I was a high-school senior. It can be solved in many ways. I will start with the simplest solution…
If the bead rotates in a circle, then the bead is rotating at constant height and is stationary relative to the reference frame “attached” to the parabola-shaped wire (as illustrated in the image). Let be the mass of the bead.
If the bead is stationary (relative to the rotating reference frame) and rotating in a circle (relative to the non-rotating inertial reference frame), then the centripetal force being applied on the bead is
The parabola exerts a “reaction” force on the bead. If that reaction force did not exist, the bead would fly away from the parabola. The reaction force is normal to the parabola-shaped wire. If it were not normal, then the reaction force projected on the parabola would result in a tangential force, which would result in tangential acceleration, and the bead would therefore not be stationary relative to the parabola. The slope of the parabola at point is . If we let
be the tangent of the angle defined by the intersection of the tangent line at point with the -axis, then it can easily be seen that the angle that the reaction force exerted on the bead and the -axis define is . If we project the normal reaction force on the vertical and horizontal axes of the rotating reference frame, we get
We have two forces being applied on the bead: the gravitational force , and the normal reaction force . Since the bead is rotating at constant height, then the sum of the vertical components of these two forces must be zero
And since , we get the norm of the reaction force
Since the bead is rotating in a circle of constant radius , then there’s a normal acceleration and zero tangential acceleration. The acceleration vector will point inwards, towards the rotation axis. This normal acceleration is due to the centripetal force. Note that this centripetal force is nothing but the projection of the reaction force on the horizontal axis:
We already know that the reaction force’s norm is , and therefore
and since , we get