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While taking to a college about calculating the centre of gravity of multibody basic objects, the question was raised on how one would determine the C.O.G of a highly heterogeneous object of a given known shape and its overall weight.

Examples of such a thing may be:

  1. A block of stone with unknown sized ores of large densities. (picture marble with different metal ores and large air bubbles, holes,cracks)

  2. Mostly empty steel object filled with mechanical parts and/or electronics (computer case?).

The positions of the masses/bodies are unknown and cannot be measured to calculate using centre of mass equation such as (under the assumption that c.o.m. and c.o.g are the same of course):

$$ \frac{ \sum_i^N m_{i} \overrightarrow{x_{j,i}} }{\sum_a^b m_{i} } $$

Knowing this point would be helpful for calculating the Lagrangian to find the equation of motion of such a thing for example.

I proposed the following experimental setup of a cube to solve the problem:

image1

Such a cube could be suspended on a near friction-less rotational joint.

From here using the linearised function of a compound pendulum:

$$ L_{com}= \frac{4 \pi ^{2}I }{T^{2} m g } $$

Assuming all variables are known, by simply measuring the period of a slightly deflected cube until stillstand. One can give an approximated distance to the centre of gravity $L_{com}$

This method is repeated on all vertices points and faces such as follows:

image 1 image2

At this point, I believe that finding the length of the c.o.g. from the centre of oscillation of the 7 primary axis (see photo) of the cube. The points (if plotted in a model of the cube will converge on the overall systems centre of gravity.

An alternate would be use the plumbline method on each suspended projected 2d dimensional face of the cube.

I feel this method is also a good one, as it requires only the ability to measure the period of the oscillation (cheap), and not the forces (expensive) as suggested on wiki.

thig

Question(s)

Is this setup an appropriate way to approximate the centre of gravity of such a hypothetical object?

If not, how can this method be improved?

Bonus question, what would be another method to find the c.o.g of such a hypothetical object?

Thanks for your help!

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    $\begingroup$ The plumbline method seems simplest. Just suspend the cube from one vertex and then from another. $\endgroup$ – G. Smith Jun 16 at 17:41
  • $\begingroup$ Mmm i would tend to agree...however i feel this method wouldnt be accurate enough, as it’d require the ability to hang from each vertex seemingly exact (so the hanging of the object itself doesn’t play a significant role and also require a way to project and measure the cube against a known point. Or would you have something in mind on how to get the measurement? :) $\endgroup$ – morbo Jun 16 at 17:51
  • $\begingroup$ I don’t know what you jean by “project and measure the cube against a known point”. All you need to measure is what point on the “bottom” of the cube is directly below the suspended vertex. You could use a very thin laser beam shining upward toward the suspension point. Turn up the power and burn a dot on the cube. $\endgroup$ – G. Smith Jun 16 at 17:58
  • $\begingroup$ Ahh that would be a method...something to consider! What i meant was that a person would have to project the cube, onto a 2d plane like in the wiki article and find the cross points. But i understand your description now Thanks! $\endgroup$ – morbo Jun 16 at 18:04
  • $\begingroup$ @G.Smith, you would probably have to suspend the cube from 3 different vertices to locate the C.G. in 3 space. $\endgroup$ – David White Jun 17 at 0:40
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You don't need the "frictionless rotating joints" to suspend the cube. Just support it on three (or more) load cells that will measure the reaction forces. By taking moments you can find a vertical line which passes through the COM of the cube.

Rotate the cube to a different orientation and repeat. Now you have two lines which intersect at the COM.

That is a practical method used in real life. If the object is an irregular shape, it may be more practical to suspend it from strings or ropes and measure the tension in each, instead of standing it on supports.

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  • $\begingroup$ Your method I infact mentioned in my link of "the forces" Namely, measuring the tension of the strings, or measuring the load on the cells. But it is expensive option and I believe my experiment could achieve the same for a fraction of the cost... It is mostly definitely a method however. Thanks. :) $\endgroup$ – morbo Jun 16 at 19:52

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