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A smooth wire has instantaneous shape $z=f(r)$ in cylindrical coordinates and rotates at constant angular velocity $\dot{\theta }=\omega$. The tangent vector at fixed time is ${\mathbf{e}}_r+f^{\prime }(r){\mathbf{e}}_z$. Its dot product with the constraint force vanishes. Projecting Newton's second law along this tangent vector gives the equation in the title.

Multiplying that equation by $\dot{r}$ proves conservation ofThis is a conserved quantity in the rotating description; the motor can exchange ordinary mechanical energy with the bead. A relative equilibrium at $a>0$ satisfies $U^{\prime }(a)=g{f}^{\prime }(a)-{\omega }^{2}a=0$. Its linearized equation is $(1+f^{\prime }(a{)}^2)\ddot{ϵ}=-U^{\prime \prime }(a)ϵ$. A strict local minimum with $U^{\prime \prime }(a)>0$ gives Lyapunov stability, while $U^{\prime \prime }(a)<0$ gives an unstable equilibrium. If $U^{\prime \prime }(a)=0$, higher orders are needed.

In cylindrical coordinates,Differentiating the position vector $\mathbf{r}=r{\mathbf{e}}_r+z{\mathbf{e}}_z$ twice gives the velocity and accelerationThis derives acceleration in cylindrical coordinates rather than assuming it.

At a fixed time, the wire has parametrization $r\mapsto r{\mathbf{e}}_r+f(r){\mathbf{e}}_z$. Its derivative is the nonzero tangent vectorA smooth wire exerts a constraint force $\mathbf{R}$ with $\mathbf{R}\cdot \mathbf{t}=0$. Because $g$ is the gravitational acceleration, the gravitational force is $-mg{\mathbf{e}}_z$; the printed $-g{\mathbf{e}}_z$ is interpreted per unit mass. Projecting Newton's second law, $m\ddot{\mathbf{r}}=\mathbf{R}-mg{\mathbf{e}}_z$, along $\mathbf{t}$ eliminates $\mathbf{R}$:The chain rule gives $\ddot{z}=f^{\prime }(r)\ddot{r}+f^{\prime \prime }(r){\dot{r}}^2$. Rearrangement yieldsThis is the governing equation for a bead on a uniformly rotating wire. Its azimuthal acceleration contributes nothing to the projection because ${\mathbf{e}}_{\theta }\cdot \mathbf{t}=0$.

In cylindrical coordinates, $\theta =\omega t$ and $z=r^2/(2a)$. ThusThe acceleration isA tangent vector along the wire is $\mathbf{t}={\mathbf{e}}_r+(r/a){\mathbf{e}}_z$. Because the wire is smooth, its constraint force is orthogonal to $\mathbf{t}$. Taking the dot product of Newton's second law with $\mathbf{t}$ eliminates that force; gravitational weight contributes $-mgr/a$. HenceMultiplying by $a^2/m$ gives