Why does spreading your legs improve stability if it lowers center of gravity?

[godingly]

Dear forum,
while standing, spreading your legs helps your stability because you have a wider base.
but doesn't spreading your legs lowers your center of gravity, thus shortening the distance (r) from your center of gravity to the ground, and therefore lowering your moment of inertia = making you easier to rotate?

The two seems to me to counteract. is that right?

 

[mathman]

The two effects, lowering c.g. and moment of inertial around a vertical axis (for rotation) are perpendicular, so changing one doesn't effect the other.

 

[jbriggs444]

Dear forum,
while standing, spreading your legs helps your stability because you have a wider base.
but doesn't spreading your legs lowers your center of gravity, thus shortening the distance (r) from your center of gravity to the ground, and therefore lowering your moment of inertia = making you easier to rotate?

The two seems to me to counteract. is that right?

The effect of spreading your legs is static. If someone is trying to tip you over toward the right, the fact that your right foot is farther right means that 1) they have farther to push to get you past the tipping point and 2) gravity has a longer moment arm to resist the tipping torque.

The effect of lowing your center of gravity is dynamic. If you assumed that the axis of rotation were on the ground below your center of gravity then widening your stance and the resultant lowering of your center of gravity could reduce your moment of inertia. But that is not where the relevant axis of rotation is located. If someone is trying to tip you over to the right, your right foot is the relevant axis of rotation. Adopting a wider stance does not move your right foot much closer to your center of gravity.

 

[mathman]

Spreading legs does affect your moment because part of you is moving away from the axis, not because you are lowering your body. Think of figure skaters moving arms in and out. the center of gravity is unaffected, but the moment changes, so they spin faster when the arms are closer to their bodies.