The bucket of water in the loop-the-loop

Problem

On a loop-the-loop a bucket of water is swung round a vertical circle. Why doesn’t the water fall out when the bucket is upside down at the top? Identify the forces on the water at the highest point.

Solution

Forces on the water at the highest point. When the bucket is upside down at the top of the circle, only two forces act on the water, both directed downward, i.e. towards the centre of the circle:

• the weight of the water $mg$;
• the normal force $N$ with which the bottom of the bucket presses on the water (the bottom is above the water and pushes downward).

Why it doesn’t fall. The water does not “fall” because it is travelling in a circle: it needs a net centripetal force directed towards the centre (downward, at that point). Newton’s second law at the highest point gives: $N + mg = \frac{m v^2}{r}$ All the weight, instead of making the water fall inside the bucket, is “used” as centripetal force: it serves to curve the water’s path, which follows the bucket around the circle. The water is falling (accelerating downward), but the bucket falls together with it at the same acceleration, so they do not separate.

Minimum speed. For the water to stay in contact with the bottom, $N\ge 0$ is required. The limiting case is $N = 0$: $mg = \frac{m v_\text{min}^2}{r} \;\Rightarrow\; \ev{v_\text{min} = \sqrt{g\,r}}$ If you swing fast enough ($v\ge v_\text{min}$), the weight alone is not enough to supply the required centripetal force and the bucket must push down on the water ($N>0$): the water stays stuck to the bottom. If instead $v < v_\text{min}$, the required centripetal force is smaller than the weight: the water detaches from the bottom and falls onto whoever is swinging the bucket. The mass cancels: the minimum speed depends only on $g$ and the radius $r$, not on how much water there is.

Links

Topics: Dinamica Concepts: Forza centripeta · Accelerazione centripeta · Forza normale · Forza peso · Moto circolare uniforme Skills: Diagramma di corpo libero · Applicazione delle leggi di Newton Methods: Metodo del diagramma di corpo libero Objects: Giro della morte