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What methods could be used to determine (or estimate within a reasonable margin of error) the mass of a living human's limbs, short of cutting them off? And more interestingly, how can this be done without any high tech equipment, just with the means commonly found in households?

A scale for example is allowed. An MRI isn't ;)

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Would a tub of water in which you submerge the limb, get the volume based on displaced water and multiple that by the average weight of a human body by within a reasonable margin? If not, what do you define as reasonable. 10% off ? – Hennes Feb 26 '13 at 22:47
@Hennes: yes, sounds like a possibility. Archimedes-style, so to speak. But how to measure the displaced water without flooding the bathroom? – 0xC0000022L Feb 27 '13 at 1:25
If you are going to fill it to the top: Not. Remove the limb then measure how much you put back into the tub to fill it up. Or mark the tub at two places: Initial place without limb. (e.g. half full). Then the place with limb (say 60-ish% full). Remove limb. Measure how much to fill until you reach the 60-ish% mark. But I mainly posted this as a comment because I was curious how much we may be off. – Hennes Feb 27 '13 at 18:02
up vote 4 down vote accepted

For those who upvoted me, I'm sorry. I just checked my equations again and I found that I made stupid mistakes when writing the torque equations. I made a mistake in writing the length of the lever arms of $W-W_{arm}$ and $W_{board}$, it should have been $L-x_a$ and $d-x_a$ instead of $L$ and $d$.

It turns out that the method in my previous answer doesn't work because every time we find $W_{arm}$, we won't find it alone. It always sticks together with $L$ as one $W_{arm}L$. At best we can only get $W_{arm}L$ as a single variable. So unless we conduct another measurement using a different method that has nothing to do with measuring center of mass, we won't be able to find $W_{arm}$. Even worse, even if we come up with other experiment of locating the center of mass or anything related to torque with various body orientations. We will always end up with $W_{arm}$ sticking with another quantity with dimension of length. And adding the number of measurements also won't help. So we have to give up trying to find $W$ with torque method and be satisfied with $W\times[Length]$. We have to figure out another experiment to find $W\times F(Length)$. The only mechanical experiment that I can think of is an experiment involving centrifugal force, since it's a dynamical experiment it's hard to measure. So I think perhaps Hennes' method is better. But if we still insist to use mechanical method here is one way to do it:

Let's say we want to calculate the weight of someone's arm. First, place a thin board horizontally and support it with a pivot and a weigh scale. Ask that guy to stand up on it but with his hands oriented straight horizontally.

enter image description here

Where $N$ is the reading of the scale times the gravitational acceleration $g$. The balance in torque gives


All quantities involved in two equations above can be measured using a scale and a meter stick except $W_{arm}$ and $L$.

Now for the second experiment, we need two high speed camera and a scale with high refresh rate. Hang a paper above the guy's head and ask him to swing his arms quickly while keeping them straight until he hits the paper. While he is doing that, record the movement of the hand using one camera and record the reading of the scale using another camera. The angular velocity $\omega$ of the hand just before hitting the paper can be obtained from the video recorded by the first camera. Angular velocity is much easier to calculate than velocity since we don't need to worry about parallax so much, but still it's not an easy task. Then we can also obtain the normal force at the instant the hand hits the paper from the second video from the reading of the scale when the voice of hitting paper is present. Let's say the reading is $N'$

enter image description here

$\Sigma F_y$ at the moment when the hand hits the paper gives

$N'=W-W_{arm}+W_{arm}(1+\frac{\omega^2 L}{g})$

Now we can substitute $L$ from one equation to the other equation to obtain $W_{arm}$. Note that $W_{arm}$ that we get is the mass of two arms, so we need to divide the final result by a factor of two to get the mass of an arm. And we can also measure the mass of a leg using the similar method.

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That's one creative method. Thanks for the explanations and the drawings! – 0xC0000022L Feb 27 '13 at 18:07
@0xC0000022L Hey I made stupid mistakes in my previous answer, sorry for that. I've updated my answer with an "uglier" one. – Emitabsorb Mar 2 '13 at 2:40

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