Question: What was the calculation used to stop the gunge from falling on Hannah's head at the start of the first lecture?

Dominique Sleet answered on 6 Jan 2020:

The gunge drop at the start of the first lecture was a great demonstration of gravity, friction and something called the conservation of angular momentum. To calculate how far the gunge would fall, there were a few things we needed to know: the weight of the gunge, the small weight at the other end of the rope, and the horizontal distance of the rope from the smaller weight to the pole around which the rope would later wrap around.

You can calculate the distance that the gunge would drop using the following equation:
2/3(horizontal rope distance) x √(small mass/large mass)

Looking at the advice of lots of other experts, it appears that this experiment works best when the heavy weight is 14 x times that of the smaller weight, so we made sure that we had the right ratio of weight between our two different objects.

But how does it work?
Well when you cut the rope, the smaller weight falls down due to gravity but it also gets pulled sideways with the rope. The causes the smaller weight to swing like a pendulum but a pendulum that is getting shorter with time. When something is rotating around a point, and you move its mass towards the centre of rotation, it will rotate faster. This is because, as the radius of the circle reduces, the distance around the circle that the mass has to travel, decreases – so it takes less time, and the mass rotates quicker. This is the idea behind the conservation of angular momentum.
This means that the smaller weight can quickly wrap itself around the pole and the friction between the rope and the pole is enough to stop the gunge falling.

However even with this maths knowledge, we made sure to do lots of testing beforehand as there’s lots of variables that are hard to factor into the equations. Such as the elasticity of the rope or the friction between the pole and the rope. On one of our first tests of the demonstration, we initially had planned to have Hannah stood with spikes either side of her head so that if the gunge fell too far it would be spiked and splatter everywhere. Unfortunately we hadn’t taken the elasticity of the rope into account and the rope jerked and hit the spikes sending gunge everywhere. Luckily we had the good sense to not have Hannah underneath the gunge for our first attempt but it did make Hannah a little bit nervous!