And what happened then?

Well, in Who-ville they say

That the Grinch’s small heart

Grew three sizes that day.

And then the true meaning

Of Christmas came through

And the Grinch found the strength

Of ten Grinches, plus two

– Dr. Seuss’s How the Grinch Stole Christmas

It’s nearly Christmas, so SteelyKid keeps demanding to watch the two classic Christmas specials we have recorded, Dr. Seuss’s How the Grinch Stole Christmas and Rudolph the Red-Nosed Reindeer. Watching these over and over again, my thoughts naturally turn to physics, and what sort of physics you could do with these shows.

The most obvious possibility is suggested by the lines above. As you no doubt remember if you’ve seen the cartoon, the Grinch steals all the Christmas trappings from the Whos down in Who-ville, loads it on a sled, and drives it ten thousand feet up the side of Mt. Crumpet, to dump it. When he hears the Whos singing their Christmas song even without their material goods, he has a change of heart, and saves the sled from falling off the cliff, using his new-found strength:

i-db183aad3a4e5fd6b79ba9f09200c41f-grinch_lifting.png

So, just how strong is the Grinch, to lift all that?

The actual lifting shot is from a funny angle, so it’s hard to work with, but there’s a cleaner shot while he’s driving up the mountain:

i-ae1d1b6e379e821b829a694dc6e7d048-grinch_driving.png

Now, the Grinch positively towers over Cindy Lou Who (who is no more than two), so let’s say he’s my height, roughly 2m tall. He’s approximately 52 pixels high in the original screen capture of that image, which works out to about 0.04m/pixel. Using that as the scale, the sack he’s standing on is 6.6m high, and the sled full of loot is 9.6m long. It’s hard to estimate the depth of the sled, but let’s say it’s roughly 4.5 m deep, a bit less than half as wide as it is long.

Treating the sacks as a giant rectangular solid, then, we would have a volume of 285 cubic meters of stuff. To convert this into a mass, we need an estimate of the density; the simple and easy density figure to remember is that water has a density of 1 g/cm3, or 1000 kg/m3. So, at the density of water, the Grinch’s sled has a mass of 285,000 kg.

Of course, water’s pretty heavy, and a lot of what’s in those sacks is considerable lighter, so let’s guess and average density of about a third that of water, and call it 100,000 kg total. Since he’s lifting that with the strength of twelve Grinches, that means a single Grinch could lift 8,333 kg. That’s around 32 times the clean and jerk lift world record, so you do not want to mess with a Grinch.

(Of course, the one in that relationship that you really don’t want to mess with is the Grinch’s faithful dog, Max, who pulls that whole sled ten thousand feet up the side of Mt. Crumpet, all by himself… Max rules.)

Given this, we can also answer a second question, namely, just how much do the Whos like Christmas? It says that “Every Who down in Who-ville liked Christmas a lot,” but that’s not very quantitative. Given the video evidence, though, we can quantify this. The population of Who-ville, the tall and the small, is exactly 33 Whos, as we can see when they’re singing:

i-6a8365da47646339a9ba430f27a82529-whos_singing.png

With 100,000 kg of Christmas trappings for all of Who-ville, that works out to 3,030 kg of Christmas gear for each and every Who. So, when they say they like Christmas a lot, they mean they like it a lot.

So, there’s your incredibly dorky analysis for this Christmas. Fah-who foraze, da-who doraze, and all that.

Comments

  1. #1 Marry Me, Mindy
    December 13, 2011

    With 100,000 kg of Christmas trappings for all of Who-ville, that works out to 3,030 kg of Christmas gear for each and every Who. So, when they say they like Christmas a lot, they mean they like it a lot.

    Alternatively, 8.6 cubic meters of stuff when packed into bags (to eliminate the density question).

    That means a pile 10 ft by 10 ft that is 3 feet deep or so. Come to think about it, that’s not so far off the kids’ haul…

  2. #2 Glenn
    December 13, 2011

    In other questions, how hot is Heat Miser’s touch to be able to melt the shovel that’s in his clutch?

  3. #3 Tim Eisele
    December 13, 2011

    Ah, but . . .

    Are these the same Whos as appeared in _Horton_Hears_A_Who_”? If they are, then an average Who is only about a micrometer tall (if that). And with the cube/square law being what it is, they would be able to perform incredible weightlifting feats similar to what we see from ants and other small insects.

    Of course, that would introduce a whole host of other questions, like how the snowflakes are still a lot smaller than an average Who. But I’ll leave that for someone else to explain away.

  4. #4 Anonymous Coward
    December 13, 2011

    What Tim Eisele said!

  5. #5 Calli Arcale
    December 13, 2011

    Fah-who foraze, da-who doraze, and all that.

    Thanks a lot, I now have that stuck in my head. :-P

  6. #6 Art
    December 13, 2011

    I think everyone can agree that the Grinch and Max are clearly the heroes of the story. The Grinch is like Judas, the hero of the Jesus mythology and the protagonist that has to do the evil deed to advance the higher truth, while risking his own vilification by the uninformed. Max is clearly the Madonna analog and the model of steadfast loyalty, fortitude, and strength of will.

    Drop the assumed height of the Grinch to five millimeters and the whole thing seems to work out from a strength of material perspective.

  7. #7 rehana
    December 13, 2011

    But are these the same Whos that Horton heard? That would make them, and the Grinch, a lot smaller.

  8. #8 Chad Orzel
    December 13, 2011

    On the scaling issue, the height of Mt. Crumpet is clearly given as 10,000 feet, and the Who houses seem more or less proportionate to that, so either they’re using Who-feet as a unit of measure (as opposed to the US foot), or Horton heard a different pack of Whos. I’m choosing to run with the latter.

  9. #9 Marry Me, Mindy
    December 13, 2011

    BTW, you need to work on your sig figs.

    “About a third” of 284,000 Kg gets to about 100,000 Kg, sure. But to go to 8333 Kg/grinch? Blah

    And 3030 Kg/Who? Double blah.

  10. #10 Raj Dhuwalia
    December 13, 2011

    A 3-size Grinch heart increase led to a 12-fold increase in Grinch strength. Let’s speculate that Grinch heart volume is proportional to Grinch strength, and that Grinch heart size increases are isotropic (i.e. proportional in all 3 dimensions). That means a 3-size jump makes a Grinch heart ~2.29 times larger. So a 1-size increase would have given him the strength of ~2.9 Grinches, and a 2-size increase the strength of ~6.4 Grinches. On the other hand, a 1-size decrease would have reduced him to less than 5% the strength of an ordinary Grinch. Just something to bear in mind in case you run into a Grinch in a dark Whoville alley, armed only with a Grinch-heart-size meter.

  11. #11 Kate Nepveu
    December 13, 2011

    OT: Raj, are you married to a friend of mine? (I have no idea how common that name is.)

  12. #12 Darren
    December 13, 2011

    10Grinch + 2 =/= 12Grinch.

  13. #13 Neil Bates
    December 14, 2011

    I “can’t help” bringing up the astonishing (to normal people) rise of Newt Gingrinch (hah, he’s famous enough that my spellchecker suggested him, also “chagrining”) in the Republican primary race – no I wouldn’t distract here merely because of his name, but naturally from his perspectives, about reducing social programs and getting poor kids to work as janitors in schools for their lunch money etc. Well, it is truly ironic and even pitiful, that folks expressing militant support for “family values” and “character counts” would be so supportive of a serial adulterer. (In all fairness, a big chunk of them aren’t, and clearly say so. Good for them, as consistency.)

    Sure, I can accept forgiving someone like that if they strayed and came back. But Newt is currently married to the woman he started an affair with against his second wife. That is like keeping what you stole and still wanting forgiveness. Furthermore, if Newt wins the Presidency, she will be our “First Lady” – so they’d make a couple of continued adulterers in our White House. This is worse than the President alone indulging in indiscretions, then giving them up later – which was bad enough. (Again, I only note this because of the name and Chad’s interest in the astonishing politics of the Right.)

  14. #14 Pat
    December 14, 2011

    I’m like my grandson! The Grinch is too scarey!

  15. #15 Joe
    December 14, 2011

    Chad (post 8), I think that the former is more in line with the story. While perhaps not the self-same Whos whom Horton heard, they are certainly of the same dust abiding species. The whole of the story takes place on an individual snow flake (as dust particles nucleate raindrops and ultimately snowflakes). But all of the reasoning done here is proportional reasoning, so it gives an answer that is merely scaled up to human proportions for understanding.

  16. #16 Isabel H
    December 14, 2011

    That’s probably one of the coolest things I’ve ever read. I’m really not that great at math not to mention that my physics education only consists of a quarter of the year way back in eighth grade, but I do buy your conclusion and I won’t question it. That’s amazing that you can just look at a picture from a movie and estimate the weight of what’s the Grinch’s Santa bag is just by examining the pixels and the context clues then doing some scaling. I don’t think it’s dorky at all, not even a waste of time, but a really cool skill and I’m jealous! Good work!

  17. #17 Tom Renbarger
    December 15, 2011

    100,000 kg, eh? So did the Grinch steal Christmas, or Festivus?

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