A Third Thought About Falling

I’m surprised my father let me get away with it. I assume that he was just being courteous and letting me get to my next points in studying whether a box is easier to tip over by pushing from the center of one of its edges or by pushing from its corner. Or he’s figured it’s too much bother to write a response; he’s been living his computer life on an iPod for a long while now and I can’t figure how he types at any length on that.

Here’s what I’m surprised he let me get away with. Last time I reasoned that the moment of inertia for making a box pivot on its corner — as needed to tip it over by pushing from the far diagonal corner — is higher than the moment of inertia for making a box pivot along one edge. And the moment of inertia is, like the inertia, a measure of how hard it is to make something start rotating. A higher moment of inertia is something harder to set rotating.

However, moment of inertia isn’t everything. What we’re really interested in is the angular acceleration, which — like the acceleration of something in a straight line — is how fast it starts rolling faster. A higher angular acceleration for a given force is going to feel like it’s rolling faster. And what matters in this angular acceleration isn’t the force by itself but the torque applied, and that’s what I expected my father to point out.

The torque is, at least as far as a physics problem cares, the distance between the axis of rotation and the point where the force is applied, times the amount of force being applied to make the object rotate. That sounds a little obscure or worse circular, but it’s put that way because the force pressing into an object might be pointing directly towards the center of mass — a radial force — or it might be pointing perpendicular to that — rotational — and the radial force will make the object move or at least break but not rotate, and the rotational force will make it spin. The farther you can get from the axis of rotation the faster you can make something spin, or, like I noticed as a kid, it’s easier to push the door open the farther you get from the hinges.

Here’s why this matters. If we’re rolling the box by its edge, and if it’s a cube where each edge is a on a side, then the farthest away we can get from the point of rotation is \sqrt{2}a. If we’re rolling the box by the corner, then the farthest the force we apply can be from the point of rotation — the far diagonal corner — is \sqrt{3}a. Couldn’t this extra bit of leverage overcome the greater moment of inertia?