Since we are still under the Halloween week, I am using the pumpkin theme as a start for the fun-physics letter of today. I would have loved to post this one the last Sunday, but it would have been too much annoying you after the previous one… so, please forgive me for the slightly-out-of-date elaboration. 

You may know that there are a variety of pumpkins (Cucurbita maxima) that can grow to incredible sizes: meters in diameter and tons in weight. We saw one of these with Olga at the Famiflora garden shop a few days ago (photo enclosed). This stimulated in my mind some dimensional analysis.

The growth rate of the giant pumpkin is extremely fast. The fruit must grow to 1000 kg in about two months, that is some 10 kg per day. these materials and nutrients have to make it through the not-so-thick pumpkin stem, with a cross section of a few cm2. This translates into a flow of about 0.1 cm-2 s-1. To make a comparison, one can look at the fastest growing living organism on Earth, the giant sequoia. The average sequoia has an age of about 100-200 years, height around 50 m and a volume of order 1000 m3, that is one million litres. The average growth is therefore in the range of 10,000 litres/year, or about 1 liter/hour. With a cross section at the base of more than 150 square meters, the climbing velocity of sap and nutrients of this monster is just 4-5 orders of magnitude smaller than in the giant pumpkin!

However, Cinderella would not be really happy to learn that her wonderfully fast-growing pumpkin at this size could never look as beautiful as she hopes… actually, all giant pumpkins are squashed to a sort of deflated, hollow ellipsoid. I looked into some data (there are wonderful resources on the internet, such as: D. Langevin, How-to-Grow World Class Giant Pumpkins, Annedawn Publishing, 1993), to make a fit of the height H of the fruit as a function of the width L. It turns out that the ratio H/L approximately follows a decreasing exponential curve (actually, the more accurate function looks like H/L=[1+exp(-t/T)]/2, that is, the height gets to half the width over a growth time of order T).

Then, my reasoning goes, that the mass M should increase as some power b of growth time M=ktb (with possibly b<1 by looking at the growth data accumulated by our meticulous farmers). By eliminating the time between the two equations, one gets a non-linear equation that tells how the ratio H/L roughly scales with the increasing mass. By looking at some plots on my Mac, I realised that the actual value of the exponent b does not make a big difference, unless it becomes very very small, therefore one can take it to be 1 (mass increasing linearly with time) or even 0.5 (parabolic growth), however the results are qualitatively the same. 

The key in deciding how much, and most importantly when, your pumpkin starts deflating from a nice spheroid into an unusable (at least from Cinderella’s point of view) bloated shape is the shell thickness, d. Actually, as I said above, one must see the pumpkin as a hollow fruit (that’s all the fun in making a Jack-o’-Lantern :)), with a shell thickness d, for which obviously we cannot use any of the classical mechanics approximations of thin shells etc. This is a VERY thick shell, which contains all the interesting food of the pumpkin!

Then, I made a plot to see at which value the H/L ratio starts declining, let us say by 10%. Around this value, we may think that the weight of the mass should initiate the collapse of the sphere (of course, one can adjust this value, and try with more or less different ones). The plot finds the “critical” size Lc as a function of the ratio d/L and, magically, I find that Lc scales as L0(d/L)2, with very little differences upon comparing various mass-laws (that is, different growth coefficients 0.2<b<1).

At this point, some dimensional analysis is in order. One can define a dimensionless ratio, (rgL)/E, namely the ratio of body forces (density times gravity times size) to elastic forces (E is the Young’s modulus of the delicious pumpkin orange matter). We can imagine that at (and above) some critical length Lc, the body forces are equal (and overwhelm) the elastic material resistance, and the pumpkin starts collapsing. In mathematical terms, that would be rgLcE.

Now, by replacing Lc with the scaling ratio above, we get (r gL0)/E ≥ (d/L)2

Let us plug in some numbers: L0 is the size at start of the gigantic growth, let us say around 20 cm; r is the density of pumpkin tissue, we can take about 2000 kg/m3g is the gravity constant, 9.81 kg m/s2E is the Young’s modulus, that some weird people have tried to measure [Shirmohammadi & Yarlagadda, Materials Today Proc. 5, 11507 (2018)] and is found to be between 3-4 MPa. I get (please check my maths!) about d/L ≥ 0.08. Since all pumpkins I ever had the chance to cut open have a d/L much larger than this, the collapse seems inevitable above some size L … Poor Cinderella, she must find a more suitable fruit for her purpose. 

However, now that I think about it, I did not include the strain rate in this simple analysis: the giant pumpkin grows in 2 months, while Cinderella’s apparently grows in just a few seconds. Going from a few cm to meters means a strain rate of order 100 s-1. In real materials such as metallic alloys or bulkpolymers, such strain rates develop extensive cavitation. Maybe… that was the secret of the Fairy Godmother!!! If enough cavitation develops, the magic pumpkin could grow and grow and keep a thickness below the critical d/L

Fairies actually know better materials science…. Bibbidi – bobbidi – Booooo!

Cinderella and her giant pumpkin

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