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What is the physics explanation for why an object with a larger base is more "stable" or more able to "balance"? For example - two pyramids, one on its tip (less stable) VS one on its base (more stable).

Also, which area of physics deals with this issue of stability, and which textbooks or monographs are good readings for the theories that deal with this?

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It has to do with angular force balance [torque = R * force(R)] around the center of mass of a rigid body.

Those objects that have wider bases have, by definition, larger R at the base to compensate for external torque applied.

Those objects that have narrower bases have smaller R at the base to compensate for torque applied, even torque applied by the body's mass itself once the body's center of mass has moved beyond R (the "tipping point").

    _____________________
   /                     \
  /                       \
 /            c            \
(         wide mass         )
              o------------->R
=============================
   _______________________
   \                     /
    \                   /
     \        c--->    /
      \               /
       \             /
        \           /
         \         /
           (narrow)
              o--->R
=============================

where c is center of mass.

The center of mass is lower in the body with wider base, so it cannot move as much to higher R (to "tipping point") as narrow body can when body pivots on base.

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With a large base, the force from the supporting surface can shift further away from a point directly below the center of gravity when necessary to oppose any external tipping torque. Any good physics text has a section on “Statics”.

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