Possibly a tough Moment of Inertia question?

Question

Parallel axis theorem is I = Io + Md^2

So say I have two spheres, equal weight and size, and I have a massless rod to hang them on. In one case I put the spheres on opposite ends and rotate about the middle of the rod -- it's really easy. In the second case I put both spheres on one end and try to rotate about the same point on the rod, it's much harder. This makes sense intuitively...

However, with a d-squared term, the moment of inertia shouldn't depend on where I put the spheres:
Case 1: I-total = Isphere + M(-D)^2 + Isphere + M(D)^2
Case 2: I-total = 2 * ( Isphere + M*(D)^2 ), which is the same.

But why, if the MOI is the same for both cases, is the balanced rod much easier to rotate?

It makes sense in my head, but not mathematically...

Thanks!

Answer 1

It's the strong desire of the doubled weights to hang directly below your hand. Since you're only interested in comparing the torque needed to accelerate the rotation, try it with the rod vertical (and the doubled weights downward), and a small back-and-forth motion. Use small rotation angles so the restoring torque doesn't get so large that it interferes with sensing the oscillatory accelerating torque. The two setups should seem more similar then. However you still have the problem of the unbalanced mass rotational acceleration producing a lateral force on your hand. That can't be changed unless you rig the rod on an axle mounted in some kind of rigidly-grounded pivot or hole.