I have a statics problem.

Here is a model for a table:

  • a circle of radius 1.5 ft
  • a pole in the center
  • a base in the shape of an X.

I place a weight on the table on the edge. How much weight can the table support before it tips over?

This is from an actual thing I am trying to build. The point mass is in fact a coffee mug (with coffee in it), the flat disc is in fact 1 inches thick. The 3 feet long pole is made of steel (and it's not quite centered). And I'm concerned the table might tip over (so far it hasn't).

Any advice how to formulate this problem more accurately? Certainly, a point mass approximation loses information. I'll try to have a picture later.

This problem should depend on a few parameters, the size of the $\times$ and the radius $R$ of the table, and the height $h$ of the table, the mass $m$ of the mug (could be very heavy $\gg 100kg$ or something absurd) and the location $r < R$ of the mug.

I'm very interested in the stability condition. This table is quite stable as long as the coffee mug is light or the $\times$ is large or maybe even if the pole is heavy...

enter image description here

  • $\begingroup$ The combined center of mass needs to be within the supports of the base. $\endgroup$ – ja72 May 1 '18 at 15:28

the table will not trip even for larger weights. since the weight of the object and the disc is exerted only on the X shaped base, unless the base is strong this will be stable, provided if the base of the table is also strongly fixed to the floor. i.e., it's just like keeping the disc on the floor and adding some weight to it. there is also the other factor, which is center of mass. If the center of mass of the combined system does not go out of the base then the system will be system will be stable.


The information given so far is insufficient to quantitatively answer your question. It would be necessary to know, a) what are the length measures of x-shaped basis; b) how far from the center is the pole; c) and how much does each element weight: the disk, the pole, the basis.

However, a rigid object does not tip over as long as its center of mass lies above its basis on the ground. Your table will not tip over if the rectangle formed by the endpoints of its x-shaped basis covers the projection of your circular table on the ground. This is because the vertical projection of the center of mass on the ground is within the projection of the boundaries of the disk.

In the crummy image, the vertical projection of the center of mass is the green line, the projection of the disk is the blue line, and the basis is the red line. As long as the point where the green line touches the ground is inside the red-bounded area, then your table will not tip. However, without putting in the actual numbers, one way to make sure that the table will not tip is if the basis covers the projection of the whole disk.

enter image description here

  • 2
    $\begingroup$ In theory, larger weights will cause the central rod to bend moving the center of mass outside the base area (potentially). $\endgroup$ – ja72 May 1 '18 at 18:14

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