What is Arnold made of?

Of course I am talking about Arnold Schwarzenegger. After looking at how many bullets he carries in Commando, I remembered this scene (also from Commando) (warning: maybe some not great language and some killing. You have been warned)

If you don't want to watch that clip, here is a shot (sorry for the quality).

i-9eb6832f57464eb56b9410acfa0a50fb-2010-06-17_you_tube_flylord_03_s_channeljpg.jpg

Clearly Arnold is strong, but there is more than strength involved here. Oh, don't bring your "he did it with wire stuff". I am not buying that. Also, I am talking about THE Arnold - he is real. I am not talking about the character in the movie (not real). Now for some physics.

Let me look at the combo of the two men (Arnold and Sully) and look at the forces acting on this "system". Picture time:

i-e59ec3b7197e57b539e2250770de0d4d-2010-06-17_untitled_4.jpg

If I assume Arnie is holding Sully just to the point where they are both going to fall, then there would just be one force on the Arnie-Sully system from the ground. It would act on the system at the edge of the cliff. The other two forces are the gravitational force on Arnie and the gravitational force on Sully.

If they are in equilibrium (and from the movie, it looks that way), then two things must be true (for this case):

i-5de754357360383ae5833f3a01aa95a1-2010-06-17_la_te_xi_t_1_10.jpg

There are no forces in the x-direction, so that condition doesn't really matter. For the torque equation, if it is not rotating about a particular point, it will not be rotating about any point. So, you can pick any point to look at the torques. Here is another picture with some distances in it. I have replaced the stick-figures with balls.

i-62e3c55bf35e44a6bac1c99dcce677ec-2010-06-17_untitled_5.jpg

Where ra is distance from Arnold's center of mass to the edge of the cliff and rb is the distance from Sulley's center of mass to the edge of the cliff. If I use the edge of the cliff to add up the torques, I get the following two equations (oh mc is the mass of Arnold-commando and ms is the mass of Sulley)

i-ca4a12eeba14c684de68aaa7eac81c9b-2010-06-17_la_te_xi_t_1_11.jpg

If I know the mass of Sulley and the two distances, then I can solve for the mass of Arnold:

i-876d411e1bdea5e0852cdad550d034ce-2010-06-17_la_te_xi_t_1_12.jpg

Now for some values. First, Arnold's arm length. Wikipedia says that he is 1.88 m tall. Using that measurement and bing.com, I made this diagram.

i-5f9300131fba29a6bdb3adea993a91ac-2010-06-17_untitled_6.jpg

Using that diagram, I can scale the image with Sulley hanging over the cliff. This is what I get (I added the red line to show the vertical line up from the edge of the cliff).

i-31e86f97dc84b20b34c76844bf261234-2010-06-21_untitled_1.jpg

From that (and one estimate of Sulley), I get:

  • rb = 0.44 m
  • ra = 0.15 m
  • ms = 68 kg (150 pounds)

Putting these values in, I get:

i-bbca75b6b8e27a83b0a8cf1c10497226-2010-06-21_la_te_xi_t_1_11.jpg

199 kg is like 440 pounds. So? I am going to calculate Arnold's density. According to Wikipedia's Arnold page, his (obviously fake) weight is listed at 250 lbs. Clearly, whoever made this fake page used Arnold's volume and the density of a normal person (about 1000 kg/m3) to calculate an expected mass. This is what they used.

i-c14ad1b2607466438d483dcfee207176-2010-06-21_la_te_xi_t_1_12.jpg

The volume of Arnold is more difficult to disguise. So, if I assume they used his correct volume then the following should be true:

i-a418db8980c9e9036e53e204dca4f30d-2010-06-21_la_te_xi_t_1_13.jpg

Using this, I can solve for Arnold's real density.

i-aef7d10bf5e3b84eb74942839f895c3e-2010-06-21_la_te_xi_t_1_14.jpg

Using his "fake weight" of 250 pounds and his real weight of 440 pounds (since I only need the ratio, it doesn't matter if I use the ratio of weights instead of the ratio of masses), I get that Arnold's density is 1.76 times the density of a human.

A good value for the density of a human is about 1000 kg/m3 (about the same as water - humans just float). This would give Arnold a density of 1750 kg/m3. So, what is he made of? Aluminum has a density around 2700 kg/m3 and titanium has a density of 4500 kg/m3. I don't know. Maybe he has some lower density stuff along with some titanium or something. He obviously isn't pure titanium. He also is obviously not human.

i-dd0dd6368965684d6a77571add6166f4-2010-06-21_term.jpg

I knew it.

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Well done! Only problem is that his center of mass doesn't seem (to me) to be accurate in your assumption. Throughout the video, Arnold's center of mass looks like it is directly over the normal force (unless, of course, Arnold's rear-end is much more dense than the rest of him).

Well, you just proved that he isn't made of solid pure titanium. But his internal metal skeleton may be made of titanium metal, and he still could have a lower density, since the volume may be filled with air or something.

Are you so smart that it makes you blind? you can clearly see the wire off his ankle at around 55 seconds.

Oh, and before I forget it: this post is pure awesome. If my physics teacher at school would have used examples like this, I would have fallen in love with physics at a much younger age. This is great!