Dear Evil Engineer: How deep would a ball pit need to be to crush someone to death?

Dear Evil Engineer,

After the first full financial year trading as an evil limited company, I celebrated by awarding myself a bonus and buying an absolutely giant shark tank (the next year, I bought sharks for it). It has served me well in the three years since, although, as it turns out: a) sharks take a great deal of maintenance; b) my water bills have run out of control; and c) I get squeamish at the sight of people being ripped apart and eaten.

I consulted with a feng shui expert, who told me the position of my tank is creating an overabundance of chi, elevating negative emotional responses. That would explain why I pushed her in. Next, I consulted with an interior designer, who recommended I replace the shark tank with a bold and colourful statement ornament to lift my mood and enhance the space.

Therefore, I am considering keeping the huge acrylic tank and filling it with something more cheerful and lower maintenance than sharks, but similarly deadly: balls. Specifically, ball-pit balls. My tank is cylindrical, with a radius of 15m and height of 10m. Can I fill it with enough balls to crush a person to death? Please help me, Evil Engineer, I don’t know what to do with all this excess chi.

Yours,
A negative and emotional villain

Dear villain,

I applaud your decision to part with your sharks, having come to the realisation that they do not suit your lifestyle. So many villains at the start of their careers are under great pressure from the industry to prove themselves with conspicuous consumption of shark tanks, flying monkeys, scantily-clad bodyguards, and designer disfigurements (very insensitive!!!). Execution by ball pit, however, is an innovation to be encouraged.

You may be familiar with the lingering phenomenon of grain engulfment, which occurs when a person becomes trapped in a large quantity of grain and – unless immediately rescued – quickly becoming completely buried due to the suction-like action of the mass of grain. Engulfment victims can suffocate within minutes.

Although grain and balls behave differently (e.g. ball-pit balls are unlikely to cause suffocation, as packed spheres touch only at a point, allowing airflow) they can both be modelled as granular materials. One of the quirks of granular materials is that, when filling a cylindrical container, pressure at the base stops increasing around the point the level of the material reaches a height greater than the cylinder’s diameter (Janssen Effect). This is thanks to grains jamming to form supportive ‘arches’, in turn supported by friction between grains and the walls of the container. As your tank is 30m in diameter and just 10m in height however, let’s disregard this effect. Let’s also model the balls as incompressible, or ‘crush resistant’, as the marketing says.

Historical records of execution by crushing suggest that 300kg of mass applied over the cross-sectional plane of an adult human body is sufficient to guarantee death in around fifteen minutes, so let’s work with that figure. What we want to calculate is the minimum lethal depth for a person to be submerged in ball-pit balls.

Let’s consider a victim lying on their back or front – the positions for which more weight is required to exert deadly pressure – and approximate their cross-section as a rectangle 1.65m in length and 0.4m in width (area of 0.66m²). Neglecting the Janssen Effect, the weight applied to the person is directly proportional to the height of the filling and thus directly proportional to the mass of ball-pit balls in a rectangular prism  extending vertically from the person-sized base.

A standard ball-pit ball weighs 6.8g and is 6.5cm in diameter (density of 47kg/m³). You would thus need 44,117 balls to provide 300kg of mass and that lethal pressure. If it was possible to pack those balls perfectly, they would fill just 6.4m³. However, we can expect a packing density of around 0.6 for balls poured randomly into a tank; hence the actual volume containing those 44,117 balls is 10.7m³. Divide that volume by the cross-sectional area of the prone victim, and we get a minimum depth of 16.2m: around the height of five African elephants standing on top of one another.

So, it would appear that a 10m-deep tank is not enough to guarantee that the typical adult will be crushed to death after sinking to the bottom of your ball pit. However, this varies depending on factors such as the shape of the victim, friction between the balls and walls, and how the balls deviate from the ideal sphere we modelled them as. For instance, if the weight of the balls crushes the balls at the bottom flat, victims would not sink all the way to the bottom.

There is no need to purchase a new tank, however; with some adjustments, you could ensure that your victims will be lethally engulfed. Although grain would be a classic, cost-effective choice of medium, trying retaining the cheerful aesthetic of a ball pit by switching the balls for denser solid balls such as bouncy rubber balls (the budget option), marbles, or billiard balls. Of course, if you are wedded to the idea of a deadly ball pit, remember that any victim dropped inside will be unable to climb out and will eventually die of thirst at the bottom, out of sight and out of mind.

Yours,
The Evil Engineer

PS: Remember to place an auspicious crystal statue of a nine-legged frog above your viewing platform. This will give you something else to look at if you get bored with staring at the ball pit.