Analysis: Will it tip or will it slide?

Now let’s look at the tipping box experiment. Here’s an FBD of the box where it’s about to tip. Now, can you see that if the box is about to tip, the friction and normal forces will be located at the front edge of the box? That’s the trick to this analysis. Note the dimensions H and a. The idea is to find H for the box to tip, rather than slide for a given coefficient of friction mu S. That’s coefficient of static limiting friction, of course. Once you’ve got the FBD sorted, the rest is a straightforward application of equilibrium. We’ll do it in stages and you can pause the video between each stage, if you like, and have a go yourself.

First, take moment about the tipping axis. It looks like a point in this diagram. This will relate to W, the weight of the box, and P, the pushing force, without involving the forces at the tipping point. Pause and try it.

Here’s the answer.

Now we can sum forces in the horizontal direction. This gives us a relationship between F and P. Similarly, summing the forces in the vertical direction relates N and W. Pause the video and try it for yourself.

Here are the answers.

Now we relate N and F using the friction equation. Pause the video and try it.

Here’s the answer.

Here our equations. We have four unknowns. H, P, F, and N. And we have four equations. So we should be able to solve. Pause the video and try to find an expression for the height, H, in terms of the assumed known quantities a and mu S, and do this for the case when the box is just about to tip.

You should get a upon H equals mu S. With this understanding you can now move furniture across a carpeted floor without it tipping. It’s a neat analysis, but I don’t think it’s as neat as the rope around a bollard, which we will look at next.