A bucket containing water is revolved in vertical circle of radius r. To prevent the water from falling down, the period of revolution required is ( g = gravitational acceleration)

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At the highest point of the vertical circle, the forces acting on the water are its weight $m g$ downwards and the normal reaction $N$ from the bucket downwards.

The necessary centripetal force is provided by these forces:

$m g + N = \frac{m v ^{2}}{r}$

To prevent the water from falling down, the normal reaction must be greater than or equal to zero ( $N \geq 0$ ). The minimum speed $v$ is obtained when $N = 0$ :

$m g = \frac{m v ^{2}}{r} \Rightarrow v = r g$

The angular velocity $ω$ corresponding to this minimum speed is:

$ω = \frac{v}{r} = \frac{g}{r}$

The maximum period of revolution $T$ required to prevent the water from falling is:

$T = \frac{2 π}{ω} = 2 π \frac{r}{g}$

Answer: $2 π \frac{r}{g}$

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