Center of gravity & moment of inertia

[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.

 

[PJC01]

Hello,

I can confirm that spreading your legs does indeed lower your center of gravity and decrease your moment of inertia. However, the overall effect on your stability is not as simple as it may seem.

While it is true that a lower center of gravity can make it easier for an object to rotate, it also increases the object's stability. This is because a lower center of gravity means that there is a shorter lever arm for external forces to act upon, making it more difficult to tip over.

Additionally, spreading your legs creates a wider base, which increases your overall stability. This is because a wider base allows for a greater distribution of weight and a larger surface area to support your body.

In summary, while spreading your legs may decrease your moment of inertia, it also has a positive effect on your stability due to the lower center of gravity and wider base. It is important to consider both factors when discussing stability and balance. I hope this helps clarify any confusion.