You may also like

problem icon

Earth Orbit

Follow in the steps of Newton and find the path that the earth follows around the sun.

problem icon

Gravity Paths

Where will the spaceman go when he falls through these strange planetary systems?

problem icon

Motorbike Momentum

A think about the physics of a motorbike riding upside down

Whirlyball

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Andrei from Tudor Vianu National College, Bucharest, Romania and Shaun from Nottingham High School both sent in excellent solutons to this problem.
forces
After drawing the picture, I observed that the forces
(the gravitational force acting on the ball, the tension in the wire and the
centrifugal force) keep the body in equilibrium. Considering the
centrifugal force, I work in a non-inertial frame of reference,
i.e. in the frame centred on the ball, which is in an
accelerated movement in respect to Earth.

In terms of the vectors we have $$m g +  F_c + T = 0.$$

Resolving horizontally and vertically and using $F=ma$ (where $a$ is the acceleration towards the centre and $T$ is the magnitude of the tension in the string):
$$
\begin{align}
T\cos \theta &= mg\\
T\sin \theta &= ml\sin \theta \omega^2\;.
\end{align}
$$
Eliminating $T$: $$l\omega^2 \cos \theta = g$$ and hence the angle $\theta$ is $\cos^{-1}{g\over l\omega^2}$.

I know that $\cos^{-1}$ is a decreasing function in the interval of interest for the problem. As $\omega$ increases, the angle $\theta$ also increases and the whirling ball rises up, the radius of its circular path also increasing.

The ball can whirl in a circle while $\theta > 0$. So, as I explained above, to find the smallest angular velocity, I have to find the smallest angle. For $\theta \to 0$, ${g\over l\omega^2}\to 1$ and so $$\omega \to \sqrt {g\over l}.$$ This is the smallest angular velocity with which the ball could rotate in a circle.

The period of this movement is: $$T = {2\pi \over \omega} = 2\pi \sqrt {l\over g}.$$ It is interesting to observe that this is also the period of isochronal oscillations of the mathematical pendulum, i.e. the period of oscillation of a material point of mass $m$ attached to an inextensible string without mass deviated from the vertical by angles less than $5^\circ$.