Q.A merry-go-round usually consists of a central vertical pillar. At the top of it there are horizontal rods which can rotate about vertical axis. At the end of this horizontal rod there is a vertical rod fitted like an elbow joint. At the lower end of each vertical rod, there is a horse on which the rider can sit. As the merry-go-round is set into rotation, these vertical rods move away from the axle by making some angle with the vertical. The figure shows vertical section of a merry-go-round in which the 'initially vertical' rods are inclined with the vertical at θ = 37°, during rotation. Calculate the frequency of revolution of the merry-go-round. (Use g = π² m/s² and sin 37° = 0.6)

Each inclined rod with its rider behaves exactly like a conical pendulum of radius $r = H + V\sin\theta = 2.1 + 1.5\times0.6 = 3.0$ m.

A merry-go-round has a horizontal rod of length H = 2.1 m from the central axle to an elbow joint, from which a vertical rod of length V = 1.5 m hangs down to a horse; during rotation these 'initially vertical' rods swing out to make angle $\theta=37^\circ$ with the vertical (use $g=\pi^2$ m/s^2 and $\sin37^\circ=0.6$). The effective radius of the rider's circular path is $H+V\sin\theta = 2.1+1.5(0.6) = 3.0$ m. Treating the inclined rod exactly like a conical-pendulum string (tension T along the rod, $T\cos\theta=mg$, $T\sin\theta=mr\omega^2=4\pi^2mrn^2$), dividing the two equations gives $\tan\theta = 4\pi^2 r n^2/g$, which rearranges to the frequency $n=\frac{1}{2\pi}\sqrt{\frac{g\tan\theta}{r}}$, evaluating to 1/2 rev/s.

Ex.1.4: Frequency of a merry-go-round from the angle its rods incline at.

Use the conical-pendulum frequency formula directly with the EFFECTIVE geometry: $n=\dfrac{1}{2\pi}\sqrt{\dfrac{g\tan\theta}{r}}$; with $g=\pi^2$, $n=\dfrac{1}{2}\sqrt{\dfrac{\tan37°}{3}}=\dfrac{1}{2}\sqrt{\dfrac{0.75}{3}}=\dfrac{1}{4}$ rev/s.

Taking the circle's radius as just $V\sin\theta$ (forgetting the horizontal rod length $H=2.1$ m), or using $L\sin\theta$ with $L$ = full rod length — the geometry here is an offset conical pendulum, $r = H + V\sin\theta$.