So the last piece I need for figuring out whether it’s easier to tip a box over by pushing on the middle of an edge or along one corner is to know the amount of torque applied by pushing with, presumably, the same force in both locations. Well, that’s almost the last bit. I also need to know how the torque and the moment of inertia connect together to say how fast an angular acceleration I can give the box.
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.
I admit not being perfectly satisfied with my answer, about whether a box is easier to tip over by pushing on the middle of one of its top edges or by pushing on its corner, just by looking at it from the energy both approaches need to raise the box’s center of mass above the pivot. It’s straightforward enough, but I don’t do this sort of calculation often, so maybe I’m looking at the wrong things. Can I find another, independent, line of argument? If I can, does that get to the same answer? If it does, good. If it doesn’t, then I get to wonder which line of argument I believe in more. So here’s one.
[ Huh. My statistics page says that someone came to me yesterday looking for the “mathematics behind rap music”. I don’t doubt there is such mathematics, but I’ve never written anything approaching it. I admit that despite the long intertwining of mathematics and music, and my own childhood of playing a three-quarter size violin in a way that must be characterized as “technically playing”, I don’t know anything nontrivial about the mathematics of any music. So, whoever was searching for that, I’m sorry to have disappointed you. ]
Now, let me try my first guess at saying whether it’s easier to tip the cube over by pushing along the middle of the edge or by pushing at the corner. I laid out the ground rules, and particularly, the symbols used for the size of the box (it’s of length ) and how far the center of mass (the dead center of the box) is from the edges and the corners last time around. Here’s my first thought about what has to be done to tip the box over: we have to make the box pivot on some point — along one edge, if we’re pushing on the edge; along one corner, if we’re pushing on the corner — and so make it start to roll. If we can raise the center of mass above the pivot then we can drop the box back down with some other face to the floor, which has to count as tipping the box over. If we don’t raise the center of mass we aren’t tipping the box at all, we’re just shoving it.
My brother phoned to remind me how much more generally nervous I should be about things, as well as to ask my opinion in an utterly pointless dispute he was having with his significant other. The dispute was over no stakes whatsoever and had no consequences of any practical value so I can see why it’d call for an outside expert. It’s more one of physics, but I did major in physics long ago, and it’s easier to treat mathematically anyway, and it was interesting enough that I spent the rest of the night working it out and I’m still not positive I’m unambiguously right. I could probably find out for certain with some simple experiments, but that would be precariously near trying, and so is right out. Let me set up the problem, though, since it’s interesting and should offer room for people to argue I’m completely wrong.